Answer with step-by-step explanation:
1) Inverse of [tex]f(x) = 6x -24[/tex]:
Make the function equal to y to get [tex]y=6x-24[/tex].
Now making [tex]x[/tex] the subject of the function:
[tex]6x=y+24\\\\x=\frac{y+24}{6} \\\\x=\frac{y}{6} +4[/tex]
Change back the variable [tex]y[/tex] to [tex]x[/tex] and this is the inverse.
[tex]f'(x)=\frac{x}{6} +4[/tex]
2. Inverse of [tex]g(x) = 3x^2 - 5[/tex]:
Making the function equal to y to get: [tex]y=3x^2 - 5[/tex]
Now making [tex]x[/tex] the subject of the function:
[tex]3x^2=y+5\\\\x^2=\frac{y+5}{3}[/tex]
Taking square root on both sides to get:
[tex]x=\sqrt{\frac{y+5}{3} }[/tex]
Change back the variable [tex]y[/tex] to [tex]x[/tex] and this is the inverse.
[tex]g'(x)=\sqrt{\frac{x+5}{3} }[/tex]
Answer:
1) [tex]f^{-1}(x)=\frac{x}{6}+4[/tex]
2) [tex]g^{-1}(x)=\sqrt{\frac{x+5}{3}}[/tex]
Step-by-step explanation:
1) To find the inverse of the function [tex]f(x) = 6x -24[/tex] you need to follow these steps:
- Since [tex]f(x)=y[/tex], you can rewrite the function:
[tex]y = 6x -24[/tex]
- Solve for "x":
[tex]y+24=6x\\\\x=\frac{y+24}{6}\\\\x=\frac{y}{6}+4[/tex]
- Exchange the variables:
[tex]y=\frac{x}{6}+4[/tex]
Then, the inverse is:
[tex]f^{-1}(x)=\frac{x}{6}+4[/tex]
2) To find the inverse of the function [tex]g(x) = 3x^2 - 5[/tex] you need to follow these steps:
- Since [tex]g(x)=y[/tex], you can rewrite the function:
[tex]y= 3x^2 - 5[/tex]
- Solve for "x":
[tex]y= 3x^2 - 5\\\\\frac{y+5}{3}=x^2\\\\x=\sqrt{\frac{y+5}{3}}[/tex]
- Exchange the variables:
[tex]y=\sqrt{\frac{x+5}{3}}[/tex]
Then, the inverse is:
[tex]g^{-1}(x)=\sqrt{\frac{x+5}{3}}[/tex]
What happens when you reflect a shape over the x-axis and then the y-axis. What is the one transformation that could have been performed to achieve the same result? A degree rotation.
Reflecting a shape over the x-axis and then the y-axis is equivalent to a 180-degree rotation around the origin.
When a shape is reflected over the x-axis, any point (x, y) on the shape will have its y-coordinate inverted, becoming (x, -y). If this reflected shape is then reflected over the y-axis, the x-coordinate is inverted, so (x, -y) becomes (-x, -y). The same final position of the shape could have been achieved by a single transformation: rotating the shape 180 degrees around the origin.
Mathematically, the combination of these two reflections can be represented by matrices and is equivalent to the composition of two reflection transformations. The reflection across the x-axis can be represented by a matrix that inverts the sign of the y-coordinate and reflection across the y-axis can be represented by a matrix inverting the x-coordinate. Together, these operations are equivalent to a 180-degree rotation around the origin.
d/dx (28000-50x²-400x)
Answer:
[tex]-100x - 400[/tex]
Step-by-step explanation:
The derivative of a sum is the sum of the derivatives by the sum rule, and this also extends to differences by the constant multiple rule
[tex]\dfrac{d}{dx}(28000 - 50x^2 - 400x) = \dfrac{d}{dx}(28000) - \dfrac{d}{dx}(50x^2) - \dfrac{d}{dx}(400x)[/tex]
By the constant multiple rule, we have
[tex]= \dfrac{d}{dx}(28000) - 50\dfrac{d}{dx}(x^2) - 400\dfrac{d}{dx}(x)[/tex]
The derivative of any constant is 0.
The power rule says that for any real number [tex]n[/tex], [tex]\frac{d}{dx} x^n = nx^{n-1}[/tex]. And note that [tex]x = x^1[/tex]. Thus we have
[tex]= 0 - 50(2)x - 400(1)x^0 = -100x - 400[/tex]
since [tex]x^0 = 1[/tex]
Explain how to model the multiplication of –6 and 3 on a number line
Answer:
Multiplication is basically repeating addition
Step-by-step explanation:
So, on a numberline, start at 0 then add -6 three times (since you are multiplying it by three) and you should get -18
Find the first six terms of the sequence. a1 = 4, an = an-1 + 8
[tex]\bf \begin{array}{llll} term&\stackrel{a_{n-1}+8}{value}\\ \cline{1-2} a_1&4\\ a_2&\stackrel{4+8}{12}\\ a_3&\stackrel{12+8}{20}\\ a_4&\stackrel{20+8}{28}\\ a_5&\stackrel{28+8}{36}\\ a_6&\stackrel{36+8}{44} \end{array}[/tex]
Answer:
The first 6 terms are 4,12,20,28,36,44
Step-by-step explanation:
So we have the recursive sequence
[tex]a_n=a_{n-1}+8 \text{ with } a_1=4[/tex].
If you try to dissect what this really means, it becomes easy.
Pretend [tex]a_n[/tex] is a term in your sequence.
Then [tex]a_{n-1}[/tex] is the term right before or something like [tex]a_{n+1}[/tex] means the term right after.
So it is telling us to find a term all we have to is add eight to the previous term.
So the second term [tex]a_2[/tex] is 4+8=12.
The third term is [tex]a_3[/tex] is 12+8=20.
The fourth term is [tex]a_4[/tex] is 20+8=28.
The fifth term is [tex]a_5[/tex] is 28+8=36
The sixth term is [tex]a_6[/tex] is 36+8=44.
Now sometimes it isn't that easy to see the pattern from the recursive definition of a relation. Sometimes the easiest way is to just plug in. Let's do a couple of rounds of that just to see what it looks like.
[tex]a_n=a_{n-1}+8 \text{ with } a_1=4[/tex].
[tex]a_2=a_1+8=4+8=12[/tex]
[tex]a_3=a_2+8=12+8=20[/tex]
[tex]a_4=a_3+8=20+8=28[/tex]
[tex]a_5=a_4+8=28+8=36[/tex]
[tex]a_6=a_5+8=36+8=44[/tex]
A 10 kg ball is traveling at the same speed as a 1 kg ball. Compared to the 10 kg ball, the 1 kg ball has...
Answer: The 1-kg ball has 1/10 as much kinetic energy as the 10-kg ball.
Step-by-step explanation:
Option (a) - "less momentum" - accurately describes the relationship between the momentum of the 1 kg ball compared to the 10 kg ball.
The momentum of an object is directly proportional to its mass. When comparing a 10 kg ball to a 1 kg ball traveling at the same speed, the 10 kg ball has more momentum due to its greater mass. Since momentum is determined by both mass and velocity, and the velocity is the same for both balls, the difference in momentum is solely due to the difference in mass. The 10 kg ball has ten times the mass of the 1 kg ball. Therefore, the 1 kg ball has less momentum compared to the 10 kg ball. In fact, it has one-tenth of the momentum of the 10 kg ball.
Complete question: A 10 kg ball is traveling at the same speed as a 1 kg ball. Compared with the 10 kg ball, the 1 kg ball has
a-less momentum.
b-twice momentum.
c-five times the momentum.
d-the same momentum.
help me with the work
first off, let's check what's the slope of that line through those two points anyway
[tex]\bf (\stackrel{x_1}{5}~,~\stackrel{y_1}{-2})\qquad (\stackrel{x_2}{7}~,~\stackrel{y_2}{2}) \\\\\\ slope = m\implies \cfrac{\stackrel{rise}{ y_2- y_1}}{\stackrel{run}{ x_2- x_1}}\implies \cfrac{2-(-2)}{7-5}\implies \cfrac{2+2}{7-5}\implies \cfrac{4}{2}\implies 2[/tex]
now, let's take a peek of what is the slope of that equation then
[tex]\bf -5y+kx=6-4x\implies -5y=6-4x-kx\implies -5y=6-x(4+k) \\\\\\ -5y=-x(4+k)+6\implies -5y=-(4+k)x+6\implies y=\cfrac{-(4+k)x+6}{-5}[/tex]
[tex]\bf y=\cfrac{(4+k)x-6}{5}\implies y=\stackrel{\stackrel{m}{\downarrow }}{\cfrac{(4+k)}{5}} x-\cfrac{6}{5}\qquad \impliedby \begin{array}{|c|ll} \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \stackrel{\textit{since both slopes are the same then}}{\cfrac{4+k}{5}=2\implies 4+k=10}\implies \blacktriangleright k=6 \blacktriangleleft[/tex]
5. Suppose U = {1, 2, 3, 4, 5, 6, 7, 8} is the universal set, and P = {1, 3, 5, 7}. What is p'? (1
point)
{2, 4, 6, 8)
{1, 2, 3, 4, 5, 6, 7, 8
O {1, 3, 5, 7}
{1, 3, 5, 7, 8)
Answer:
p' = {2, 4, 6, 8}.
Step-by-step explanation:
The set p' has all the elements in the universal set U that are not in set p.
p' = {2, 4, 6, 8}.
Step-by-step answer:
U = universal set (all possible members)
P = given set
P' = complement of P, i.e. contains all members in U but NOT 1,3,5,7in P.
Thus, U=P or P'.
Here,
U={1,2,3,4,5,6,7,8}
P={1,3,5,7}
so P'={2,4,6,8}
Check: P or P' ={1,3,5,7} or {2,4,6,8} = {1,2,3,4,5,6,7,8} = U good !
Consider the table of values for function f.
x: -1, 0, 1, 2, 3, 4, 5
f(x): 3.5, 4, 5, 7, 11, 19, 35
The function g is shown below:
g(x)=4x+5
In which of the following intervals is the average rate of change greater for f than for g?
A. [1,3]
B. [3,5]
C. [0,2]
D. [-1,0]
Answer:
B. [3,5]
Step-by-step explanation:
The rate of change of a function is the same as the slope between two given points from that same function,
Hence,
all we need to do is use the slope's equation, that is
[tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex]
And eval it in every couple of ordered pairs given from the table we obtain the following>
x2 y2 x1 y1 m
0 4 -1 3,5 0,5
1 5 0 4 1
2 7 1 5 2
3 11 2 7 4
4 19 3 11 8
5 35 4 19 16
The rate of change from g(x) is 4 (its slope)
Hence, the interval when the rate of change of f(x) is greater than g(x) is from x=3 to x=5
What is the range of this relation (2,-3),(-4,2),(6,2),(-5,-3),(-3,0)
Answer:
Range of this relation = -3, 2
Step-by-step explanation:
We are given the following relation and we are to find its range:
[tex](2,-3),(-4,2),(6,2),(-5,-3),(-3,0)[/tex]
The set of all the possible dependent values a relation can produce from its values of domain are called its range. In simple words, it is the list of all possible inputs (without repeating any numbers).
Therefore, the range of this relation is: -3, 2
Answer: [tex]Range:[/tex]{[tex]-3,0,2[/tex]}
Step-by-step explanation:
The range of a relation is the set of y-coordinates of the ordered pairs (These are the second numbers of each ordered pair).
In this case you have the following relation:
[tex](2,-3),(-4,2),(6,2),(-5,-3),(-3,0)[/tex]
Therefore, based on the explained bofore, you can conclude that the rsnge of the given relation is the following:
[tex]Range:[/tex]{[tex]-3,0,2[/tex]}
(Notice that you do not need to write the same number twice)
Find the limit of the function by using direct substitution. (6 points) limit as x approaches zero of quantity x squared minus three.
Answer:
[tex]\lim_{x \to 0} x^2-3=-3[/tex]
Step-by-step explanation:
This limit can be written as follows
[tex]\lim_{x \to 0} x^2-3[/tex]
Direct substitution means that we substitute in the value for x to get our limit
[tex]\lim_{x \to 0} x^2-3\\\\0^2-3\\\\-3[/tex]
[tex]\displaystyle\\\lim_{x\to 0}(x^2-3)=0^2-3=-3[/tex]
Which phrase matches the algebraic expression below?
2(x − 7) + 10
Answer:
Two times the difference of a number x and 7 plus 10.
Step-by-step explanation:
We are to write a phrase which matches the following algebraic expression below:
[tex] 2 ( x − 7 ) + 1 0 [/tex]
We can see that there is a bracket with a coefficient outside it so we can express it as:
'Two times the difference of a number x and 7'.
While + 10 can be added to it to complete the phrase:
Two times the difference of a number x and 7 plus 10.
Rachel is making bouquets she has 72 carnations and 48 roses. Each bouquet has the same number of flowers, with no flowers left over. What is the greatest number of bouquets she can make?
There are 24 bouquets making by Rachel.
What is Factor?
A number which means to break it up into numbers that can be multiplied together to get the original number.
Now, To solve this problem,
Let us first lay out all the factors of each number.
72 : 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
48 : 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
So, The greatest number of bouquets that can be made would be equal to the greatest common factor of the two numbers 72 and 48 will 24.
Hence, There are 2 bouquets making by Rachel.
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Rachel can make 24 bouquets with 72 carnations and 48 roses.
To determine the greatest number of bouquets Rachel can make with 72 carnations and 48 roses, we need to find the greatest common divisor (GCD) of 72 and 48. The GCD is the largest number that divides both 72 and 48 without leaving a remainder.
Here are the steps to find the GCD:
List the factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72List the factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48Identify the common factors: 1, 2, 3, 4, 6, 8, 12, 24The greatest common factor is 24.ANY HELP WOULD REALLY HELP ME :)
White shapes and black shapes are used in a game some of the shapes ae circles all the other shapes are squares.
The ratio of the number of white shapes to the number of black shapes is 5:11 the ratio of the number of white circles to the number of white squares 3:7
The ratio of the number of black circles to the number of black squares is 3:8 work out what fraction of all the shapes are circles.
Answer:
6/11
Step-by-step explanation:
3 white circles, 3 black circles, so six circles out of the total 11 peices
A bag contains 5 blue, 3 red, and 8 green marbles. You choose a marble, do not replace it, and then choose another one. What is the probability that both marbles are red?
Answer:
1/40
Step-by-step explanation:
A bag contains 5 blue, 3 red, and 8 green marbles
You have (5+3+8=16) marbles
P(red 1) = red/total = 3/16
You do not replace it
A bag contains 5 blue, 2 red, and 8 green marbles
You have (5+2+8=15) marbles
P(red 2nd) = red/total = 2/15
P(red 1, red 2) = P (red 1)* P (red 2) = 3/16 * 2/15
=1/40
Which statement best describes the polynomial -8x^4 ? first degree polynomial with two terms fourth degree polynomial with two terms fourth degree monomial second degree binomial
Answer:
fourth degree monomial
Step-by-step explanation:
-8x^4
There is one term so it is a monomial
The highest power is degree 4, so it would be a quartic
A ball has a surface area of 26.87 square feet. To the nearest tenth of a foot, what is the radius of the ball? Use π = 3.14.
Answer:
about 1.5 feet
Step-by-step explanation:
First we need to note the equation to find the surface area of a sphere. This is 4πr^2.
4πr^2=Surface Area
4πr^2=26.87
4(3.14)r^2=26.87 Substitute
r^2=26.87/(4*3.14) Divide
r^2=26.87/12.56 Simplify
r=√(26.87/12.56) Find square root of both sides
r≈1.5 Do the calculations
So the radius of the ball is about 1.5 feet.
Answer:1.5 ft
Step-by-step explanation:
Given
Surface area of ball=[tex]26.87 ft^2 [/tex]
And we know surface area of ball is =[tex]4\pi r^2[/tex]
Equating
[tex]26.87=4\times 3.14\times r^2 [/tex]
[tex]r^2=2.1393[/tex]
[tex]r=1.4626\approx 1.5 [/tex]
Trisha has $22.50 to spend at the movies . Her ticket is $11.50, and she gets a large popcorn and soda. If the popcorn and soda each cost the same amount and she spends all her money , how much was the popcorn?
Answer:
The popcorn was $5.50
Answer:
$5.50
Step-by-step explanation:
22.50 - 11.50 = 11
divide 11 by 2 and you get 5.5
Hope this helps!
Which equations represent the line that is parallel to 3x − 4y = 7 and passes through the point (−4, −2)? Select two options.
Options:
1. y= -3/4x+1
2. 3x-4y=-4
3.4x-3y=-3
4. y-2=-3/4(x-4)
5. y+ 2 = 3/4(x + 4)
Please help ASAP thank you ! :)
Answer:
2 and 5
Step-by-step explanation:
The slope-intercept form of a line is y=mx-b where m is slope and b is y-intercept.
The point-slope form of a line is y-y1=m(x-x1) where m is the slope and (x1,y1) is a point on the line.
The standard form a line is ax+by=c.
So anyways parallel lines have the same slope.
So if we are looking for a line parallel to 3x-4y=7 then we need to know the slope of this line so we can find the slope of our parallel line.
3x-4y=7
Goal: Put into slope-intercept form
3x-4y=7
Subtract 3x on both sides:
-4y=-3x+7
Divide both sides by -4:
[tex]y=\frac{-3}{-4}x+\frac{7}{-4}[/tex]
Simplify:
[tex]y=\frac{3}{4}x+\frac{-7}{4}[/tex]
So the slope of this line is 3/4. So our line that is parallel to this one will have this same slope.
So we know our line should be in the form of [tex]y=\frac{3}{4}x+b[/tex].
To find b we will use the point that is suppose to be on our new line here which is (x,y)=(-4,-2).
So plugging this in to solve for b now:
[tex]-2=\frac{3}{4}(-4)+b[/tex]
[tex]-2=-3+b[/tex]
[tex]3-2=b[/tex]
[tex]b=1[/tex]
so the equation of our line in slope-intercept form is [tex]y=\frac{3}{4}x+1[/tex]
So that isn't option 1 because the slope is different. That was the only option that was in slope-intercept form.
The standard form of a line is ax+by=c and we have 2 options that look like that.
So let's rearrange the line that we just found into that form.
[tex]y=\frac{3}{4}x+1[/tex]
Clear the fractions because we only want integer coefficients by multiplying both sides by 4.
This gives us:
[tex]4y=3x+4[/tex]
Subtract 3x on both sides:
[tex]-3x+4y=4[/tex]
I don't see this option either.
Multiply both sides by -1:
[tex]3x-4y=-4[/tex]
I do see this as a option. So far the only option that works is 2.
Let's look at point slope form now.
We had the point that our line went through was (x1,y1)=(-4,-2) and the slope,m, was 3/4 (we found this earlier).
y-y1=m(x-x1)
Plug in like so:
y-(-2)=3/4(x-(-4))
y+2=3/4 (x+4)
So option 5 looks good too.
Answer:
OPTION 2.
OPTION 5.
Step-by-step explanation:
The equation of the line in Slope-Intercept form is:
[tex]y=mx+b[/tex]
Where "m" is the slope and "b" is the y-intercept.
Given the line [tex]3x - 4y = 7[/tex], solve for "y":
[tex]3x - 4y = 7\\\\-4y=-3x+7\\\\y=\frac{3}{4}x-\frac{7}{4}[/tex]
The slope of this line is:
[tex]m=\frac{3}{4}[/tex]
Since the slopes of parallel lines are equal, the slope of the other line is:
[tex]m=\frac{3}{4}[/tex]
Substitute the slope and the given point into [tex]y=mx+b[/tex] and solve for "b":
[tex]-2=\frac{3}{4}(-4)+b\\\\-2+3=b\\\\b=1[/tex]
Then, the equation of this line in Slope-Intercept form is:
[tex]y=\frac{3}{4}x+1[/tex]
The equation of the line in Standard form is:
[tex]Ax+By=C[/tex]
Then, manipulating the equation [tex]y=\frac{3}{4}x+1[/tex] algebraically, we get:
[tex]y-1=\frac{3}{4}x\\\\4(y-1)=3x\\\\4y-4=3x\\\\-4=3x-4y\\\\3x-4y=-4[/tex]
An equation was used to predict the number of possible enrollments in an afterschool program for the first 6 months of the year. The actual enrollments are also listed.
Actual enrollment 55 80 95 100 115 90
Predicted enrollment 75 80 85 90 95 100
The sum of the residuals is ______.
Answer:
70
Step-by-step explanation:
75-55=20
95-85=10
100-90=10
115-95=20
100-90=10
sum of the differences is 70
Answer:
A: 10
Step-by-step explanation:
Let's add the actual enrollments all together. 55 + 80 + 95 + 100 + 115 + 90 all added together is 535. The predicted enrollments, all added together, is: 75 + 80 + 85 +90 +95 + 100 = 525.
535 - 525 = 10 residuals.
all of the following expressions have the same value except... -4/5-1/5 -(4/5+1/5) -4/5+ -1/5 -4/5-(-1/5)
Answer:
-4/5-(-1/5)
Step-by-step explanation:
-4/5-1/5 is equivalent to -4/5+-1/5 which equals -1
-(4/5+1/5) when you add the expression in the parenthesis you get 1 and when you multiply that by -1 you get -1
-4/5+ -1/5 this is like the first choice so it equals -1
-4/5-(-1/5) this expression is the equivalent to -4/5+1/5 which gives you -3/5
The last option is -3/5 while the others are -1 which it the one with the different value
All of the expressions have the same value except -4/5 - (-1/5).
Explanation:In this case, all of the expressions have the same value except -4/5 - (-1/5).
To simplify each expression:
-4/5 - 1/5 can be simplified to -5/5 which is equal to -1.-(4/5 + 1/5) can be simplified to -5/5 which is also equal to -1.-4/5 + (-1/5) can be simplified to -5/5 which is still equal to -1.Finally, -4/5 - (-1/5) can be simplified to -4/5 + 1/5. Adding these fractions gives us -3/5.Therefore, the expression that has a different value is -4/5 - (-1/5).
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Consider the quadratic equation 2x2 + 3x + 5 = 0.
The discriminant of this quadratic equation is ________ zero.
This means the quadratic equation will have ________ real solution(s) and _______ complex solution(s).
Answer:
The discriminant of this quadratic equation is less than zero.
This means the quadratic equation will have no real solution(s) and two complex solution(s).
Step-by-step explanation:
The discriminant is found using the formula b^2 - 4ac.
Therefore, the discriminant is (3)^2 - 4(2)(5), which yields -31.
Since the discriminant is negative, there are no real solutions.
Answer: less then
No
Two distinct
Step-by-step explanation:
Use the grouping method to factor this polynomial completely.
4x3 + 8x2 + 3x+6
Answer:
(x + 2)(4x^2 + 3)
Step-by-step explanation:
The first two terms factor as follows: 4x^2(x + 2).
The last two factor as follows: 3(x + 2).
Thus, (x + 2) is a factor of 4x3 + 8x2 + 3x+6:
4x^2(x + 2) + 3(x + 2), or:
(x + 2)(4x^2 + 3).
Note that 4x^2 + 3 can be factored further, but doing so yields two complex roots.
Answer: (x + 2)(4x^2 + 3)
Step-by-step explanation:
4x3 + 8x2 + 3x+6 becomes 4x^2(x + 2) + 3(x + 2) and can also be written as (x + 2)(4x^2 + 3).
----------------------
Now you know the answer as well as the formula. Hope this helps, have a BLESSED AND WONDERFUL DAY!
- Cutiepatutie ☺❀❤
Page 1:
(MC 02.03)
What set of reflections and rotations would carry rectangle ABCD onto itself?
(4 points)
1) Rotate 180°, reflect over the x-axis, reflect over the line yox
2) Reflect over the x-axis, rotate 180°, reflect over the x-axis
3) Rotate 180°, reflect over the y-axis, reflect over the line y=x
4) Reflect over the y-axis, reflect over the x-axis, rotate 180°
Answer:
4) Reflect over the y-axis, reflect over the x-axis, rotate 180°
Step-by-step explanation:
Since, the rule of 180° rotation,
[tex](x,y)\rightarrow (-x,-y)[/tex]
Rule of reflection over x-axis,
[tex](x,y)\rightarrow (x,-y)[/tex]
Rule of reflection over y-axis,
[tex](x,y)\rightarrow (-x,y)[/tex]
Rule of reflection over line y = x,
[tex](x,y)\rightarrow (y,x)[/tex]
∵ In the set of reflection,
Rotate 180°, reflect over the x-axis, reflect over the line y=x,
[tex](x,y)\rightarrow (-x,-y)\rightarrow (-x,y)\rightarrow (y,-x)[/tex]
Reflect over the x-axis, rotate 180°, reflect over the x-axis,
[tex](x,y)\rightarrow (x,-y)\rightarrow (-x,y)\rightarrow (-x,-y)[/tex]
Rotate 180°, reflect over the y-axis, reflect over the line y=x,
[tex](x,y)\rightarrow (-x,-y)\rightarrow (x,-y)\rightarrow (-y,x)[/tex]
Reflect over the y-axis, reflect over the x-axis, rotate 180°,
[tex](x,y)\rightarrow (-x,y)\rightarrow (-x,-y)\rightarrow (x,y)[/tex]
Hence, the set of reflections and rotations that would carry rectangle ABCD onto itself is,
Reflect over the y-axis, reflect over the x-axis, rotate 180°.
Option '4' is correct.
Answer:
Reflect over the y-axis, reflect over the x‒axis, rotate 180°
Step-by-step explanation:
The functions f(x) and g(x) are defined below. f(x)=sq. root x^2+12x+36 g(x)=x^3-12 Which expression is equal to f(x) · g(x)?
Answer: [tex]\bold{x^4+6x^3-12x-72}[/tex]
Step-by-step explanation:
[tex]f(x) = \sqrt{x^2+12x+36} \implies f(x)=\sqrt{(x+6)^2}\implies f(x) = x+6\\\\\\f(x)\cdot g(x)=(x+6)(x^3-12)\\.\qquad \qquad =x(x^3-12)+6(x^3-12)\\.\qquad \qquad =x^4-12x +6x^3-72\\.\qquad \qquad =\large\boxed{x^4+6x^3-12x-72}[/tex]
The expression equal to f(x)·g(x) is [tex]x^{4} + 6x^{3} - 12x - 72[/tex] .
What is the equivalent expression of f(x)·g(x) after simplification ?The expressions given are f(x) = [tex]\sqrt{x^{2}+12x+36}[/tex] and g(x) = [tex]x^{3} - 12[/tex] .
Thus we have
⇒ f(X) = [tex]\sqrt{x^{2}+12x+36}[/tex]
⇒ f(x) = [tex]\sqrt{(x+6)^{2} }[/tex]
⇒ f(x) = x + 6 .
Therefore the expression of multiplication of f(x)·g(x) is =
= [tex](x^{3} - 12)*(x + 6)[/tex]
= [tex]x^{4} + 6x^{3} - 12x - 72[/tex]
Thus, the expression equal to f(x)·g(x) is [tex]x^{4} + 6x^{3} - 12x - 72[/tex] .
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which set of three numbers could be the side lengths of an triangle 3,5,9 2,4,6 2,4,8 3,5,7
Answer:
3,5,7
Step-by-step explanation:
to find which set it is you have to add the first too like this
3+5 >7 if greater than 7 move on too this
5+7>3 if greater move on to this
3+7>5 if greater that means it is a set you can use
3,5,7 is the set of three numbers that could be the side lengths of a triangle. This can be obtained by using the triangle inequality theorem.
What is triangle inequality theorem?Triangle inequality theorem: The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
If ΔABC is a triangle with sides a, b and c, then it should satisfy the following conditions, a+b>c
b+c>a
a+c>b
Which is the required set of three numbers?
Use the triangle inequality theorem in each set,
Set 1: 3,5,93+5=8<9, does not satisfy the theorem
Set 2: 2,4,62+4=6, does not satisfy the theorem
Set 3: 2,4,82+4=6<8, does not satisfy the theorem
Set 4: 3,5,73+5=8>7
3+7=10>5
5+7=12>3, this set satisfies the theorem
Hence 3,5,7 is the set of three numbers that could be the side lengths of a triangle.
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f(x) = x^2-5
g(x) = 4x - 4
Find (f-g) (5)
Answer:
The correct answer option is: 4.
Step-by-step explanation:
We are given the following two functions and we are to find the value of [tex] ( f - g ) ( 5 ) [/tex]:
[tex]f(x) = x^2 - 5[/tex]
[tex]g(x) = 4x - 4[/tex]
Finding [tex] ( f - g ) ( x ) [/tex]:
[tex] ( f - g ) ( x ) [/tex] [tex]= (x^2-5)-(4x-4) = x^2-4x-5+4[/tex]
[tex]( f - g ) ( x ) = x^2-4x-1[/tex]
So, [tex]( f - g ) ( 5 ) = (5)^2-4(5)-1 = 4[/tex]
Which statement is true about the factorization of 30x2 +40xy+51y2
Answer:
prime
not factorable over the reals
Step-by-step explanation:
30x^2+40xy+51y^2 is not factorable (also know as prime)
because the discriminant is negative.
That is when I computed b^2-4ac, in this case 40^2-4(30)(51)=-4520.
If the discriminant is negative, then it isn't factorable over the reals.
Solve 16t2 - 96t + 48 = 0.
Answer:
the question is not written correctly 16t2 can't be used in the equation
Step-by-step explanation:
do it it correctly you need to combine the like numbers, subtract the 48 from both sides and then divide the number with the t to both sides.
We get the value of t as:
[tex]t=3+\sqrt{6}\ and\ t=3-\sqrt{6}[/tex]
Step-by-step explanation:We are asked to solve the given equation for t.
The quadratic equation in terms of the variable t is given by:
[tex]16t^2-96t+48=0[/tex]
On dividing both side of the equation by 16 we get:
[tex]t^2-6t+3=0[/tex]
Now, we know that any quadratic equation of the type:
[tex]at^2+bt+c=0[/tex] has solution as :
[tex]t=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]
Here,
[tex]a=1,\ b=-6\ and\ c=3[/tex]
i.e.
[tex]t=\dfrac{-(-6)\pm \sqrt{(-6)^2-4\times 1\times 3}}{2\times 1}\\\\t=\dfrac{6\pm \sqrt{36-12}}{2}\\\\t=\dfrac{6\pm \sqrt{24}}{2}\\\\t=\dfrac{6\pm 2\sqrt{6}}{2}\\\\t=3\pm \sqrt{6}[/tex]
i.e.
[tex]t=3+\sqrt{6}\ and\ t=3-\sqrt{6}[/tex]
What is the sum of The square root of negative 2 and the square root of negative 18
Answer:
[tex]\large\boxed{\sqrt{-2}+\sqrt{-18}=4\sqrt2\ i}[/tex]
Step-by-step explanation:
[tex]\sqrt{-1}=i\\\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\\===================\\\\\sqrt{-2}+\sqrt{-18}=\sqrt{(2)(-1)}+\sqrt{(9)(2)(-1)}\\\\=\sqrt2\cdot\sqrt{-1}+\sqrt9\cdot\sqrt2\cdot\sqrt{-1}\\\\=\sqrt2\cdot i+3\cdot\sqrt2\cdot i\\\\=i\sqrt2+3i\sqrt2=4i\sqrt2[/tex]
The sum of √-2 and √-18 is 4√2i.
What is the square root of -1?The square root of -1 is an imaginary number which is represented by i.
√-1=i
Here we have to calculate √-2+√-18
√-2+√-18
=√(-1).2+√(-1).18
=√(-1).√2+√(-1).√18
=i√2+i√18 (as √-1=i where i is imaginary number)
But √18=√(9*2)=√9*√2=3√2
(as √(ab)=√a.√b)
=i√2+i3√2
=√2(i+3i)
=√2*4i
=4√2i
Therefore the sum of √-2 and √-18 is 4√2i.
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Which best describes the graph of the function f(x)=4x ?
Answer:
F
Step-by-step explanation:
The equation of a line in slope-intercept form is
y = mx + b
where m = slope, and b = y-intercept.
This line has equation
y = 4x
When we compare y = 4x with y = mx + b, we see that m = 4, and b = 0.
This line intersects the y-axis (the y-intercept) at y = 0, which is the origin. The slope is positive, so it slopes up to the right.
Answer: F
The function f(x) = 4x is a linear function, and when graphed, it will produce a straight line with a slope of 4. The graph demonstrates the dependence of y on x and it's an example of a line graph.
Explanation:The question asks for a description of the graph of the function f(x) = 4x.
The function f(x) = 4x is a linear function, which means it creates a straight line when graphed. The number 4 in the function is the slope of the line. This value indicates that for each increase by 1 in the x value, the y value will increase by 4.
This demonstrates the dependence of y on x. So, a graph of this function would be a straight line that slopes upward from left to right, starting at the origin (0,0) and increasing by 4 in the y direction for each step of 1 in the x direction. This is a typical representation of a linear function demonstrating the relationship between two variables on a line graph.
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