To find h(x) = f(x) - g(x), we substitute and simplify the given functions. The result is h(x) = 2 × 3ˣ - 3²ˣ. This solution involves basic algebraic manipulation and exponent rules.
Let's start by defining the given functions:
f(x) = 3ˣg(x) = 3²ˣ - 3ˣWe need to find h(x) = f(x) - g(x). Substitute the expressions for f(x) and g(x):
h(x) = 3ˣ - (3²ˣ - 3ˣ)Now simplify the expression inside the parentheses:
h(x) = 3ˣ - 3²ˣ + 3ˣCombine like terms:
h(x) = 3ˣ+ 3ˣ - 3²ˣh(x) = 2 × 3ˣ - 3²ˣThus, the function h(x) = 2 × 3ˣ - 3²ˣ is found by subtracting g(x) from f(x).
There is a two-digit number whose units digit is six less than the tens digit. Four times the tens digit plus five times the units digit equal 51. Find the digits.
Answer:
The tens digit number is 9 and the ones digit number is 3.
Which system of equations below has infinitely many solutions?
y = –3x + 4 and y = –3x – 4
y = –3x + 4 and 3y = –9x + 12
y = –3x + 4 and y = -1/3x + 4
y = –3x + 4 and y = –6x + 8
Answer:
y = –3x + 4 and y = –3x – 4 no solution exists
y = –3x + 4 and 3y = –9x + 12 y=4-3x
y = –3x + 4 and y = -1/3x + 4 x=0, y=4
y = –3x + 4 and y = –6x + 8 x=4/3, y=0
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Step-by-step explanation:
Jim goes to the store and buy 6 apples for 25 cent each and 10 bananas for 10 cent each Jim has a 10% off coupon Jim will own the store $2. 25 truє σr fαlѕє
Simplify the rational expression. State any restrictions on the variable n^4-11n^2+30/ n^4-7n^2+10
The given rational expression simplifies to (n^2 - 6)/(n^2 - 2). However, n cannot be equal to ±√2 and ±√5. These are the values that would make the denominator of the original or simplified expression equal to 0.
Explanation:To simplify the given rational expression we must first factor both the numerator and the denominator. The expression is n^4 - 11n^2 + 30/ n^4 - 7n^2 + 10. We recognize this is a quadratic in terms of n^2. Hence, the numerator factors into (n^2 - 6)(n^2 - 5), and the denominator factors into (n^2 - 5)(n^2 - 2).
The simplified version of the expression is (n^2 - 6)/(n^2 - 2) but we need to take into account the restrictions for n which are that n cannot be equal to ±√2 and ±√5. These are the values that would make the denominator of the original or simplified expression equal to 0, thus undefined.
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The rational expression [tex](n^4 - 11n^2 + 30)/(n^4 - 7n^2 + 10)[/tex] simplifies to [tex](n^2 - 6)/(n^2 - 2)[/tex], with restrictions on n being ±√5 and ±√2 since these values would cause the original denominator to be zero.
To simplify the rational expression [tex]n^4 - 11n^2 + 30[/tex] divided by [tex]n^4 - 7n^2 + 10[/tex], we start by factoring both the numerator and the denominator.
Factor the numerator:
→ [tex]n^4 - 11n^2 + 30 = (n^2 - 5)(n^2 - 6)[/tex]
Factor the denominator:
→ [tex]n^4 - 7n^2 + 10 = (n^2 - 5)(n^2 - 2)[/tex]
After factoring, we can eliminate common terms.
The term ([tex]n^2 - 5[/tex]) is present in both the numerator and the denominator and can be cancelled out.
The simplified expression is [tex](n^2 - 6)/(n^2 - 2)[/tex].
We also need to state restrictions on the variable n. These restrictions come from the values that make any denominator zero, which are not allowed.
For the initial fraction, the restrictions are the values of n that make [tex]n^2 - 5[/tex] or [tex]n^2 - 2[/tex] equal to 0. Thus, n cannot be ±√5 or ±√2.
The table below shows the average temperature, in degrees Celsius, in Jacob's city over a period of five months:
Month 1 2 3 4 5
Temperature 5 7.2 9.4 11.6 13.8
Did the temperature in Jacob's city increase linearly or exponentially?
Linearly, because the table shows a constant percentage increase in temperature each month
Exponentially, because the table shows a constant percentage increase in temperature each month
Linearly, because the table shows that temperature increased by the same amount each month
Exponentially, because the table shows that temperature increased by the same amount each month
Answer:
C. Linearly, because the table shows that temperature increased by the same amount each month
Step-by-step explanation:
Which operation is performed in the derivation of the quadratic formula moving from Step 6 to Step 7? subtracting from both sides of the equation squaring both sides of the equation taking the square root of both sides of the equation taking the square root of the discriminant
Answer:
C
Step-by-step explanation:
Write an equation that involves multiplication,addition, contains a variable and has a solution of 8.
Does the point (2 3 , 2) lie on the circle that is centered at the origin and contains the point (0, -4)? Why?
Answer:
No, because it doesn't satisfy the equation of the circumference
Step-by-step explanation:
A circle is the locus of points on the plane that are equidistant from a fixed point called the center. For a circle whose center is the point
[tex]C=(a,b)[/tex]
and its radius is [tex]r[/tex], the ordinary equation of this circle is given by:
[tex](x-a)^2+(y-b)^2=r^2[/tex]
Since the circle is centered at the origin:
[tex]C=(a,b)=(0,0)\\\\Hence\\\\(x-0)^2+(y-0)^2=r^2\\\\x^2+y^2=r^2[/tex]
Now, let's find [tex]r[/tex] using the data provided. Evaluating the point (0,-4) into the equation:
[tex](0)^2+(-4)^2=r^2\\\\16=r^2\\\\r=\pm 4[/tex]
Thus the equation for the circle given by the problem is:
[tex]x^2+y^2=16[/tex]
In order to corroborate if the the point (2 3, 2) lie on the circle, we need to evaluate it into the equation and check if it satisfy the equation:
Note: I don't know what you mean with 2 3, so I will assume 3 cases:
[tex]2\hspace{3} 3=23\\2\hspace{3} 3=2*3=6\\2\hspace{3} 3=\frac{2}{3}[/tex]
First case:
[tex](23)^2+(2)^2=16\\\\533\neq16[/tex]
It doesn't satisfy the equation, therefore doesn't lie on the circle.
Second case:
[tex](6)^2+(2)^2=16\\\\40\neq16[/tex]
It doesn't satisfy the equation, therefore doesn't lie on the circle.
Third case:
[tex](\frac{2}{3} )^2+(2)^2=16\\\\\frac{40}{9} \neq16[/tex]
It doesn't satisfy the equation, therefore doesn't lie on the circle.
Final answer:
The point (3, 2) does not lie on the circle centered at the origin with a radius of 4 units because when its coordinates are plugged into the circle's equation x² + y² = 16, the result is not equal to the radius squared.
Explanation:
To determine if the point (3, 2) lies on the circle centered at the origin that contains the point (0, -4), we need to see if it satisfies the equation of the circle.
We know the radius of the circle is the distance from the center to the point (0, -4), which is 4 units since all points on a circle are equidistant from the center.
The general equation for a circle centered at the origin (0,0) is x² + y² = r², where r is the radius.
Our circle's equation is x² + y² = 16.
We plug in the coordinates of the point (3, 2) to see if it lies on the circle: 3² + 2² = 9 + 4 = 13, which is not equal to 16. Therefore, the point (3, 2) does not lie on the given circle.
Which expression is equivalent to a(^8)^4? a a2 b a4 c a12 d a32
We have been given the expression [tex] (a^8)^4 [/tex].
To solve this problem we will use one of the rules of the exponents called the Power Rule.
The power rule states that:
[tex] (x^m)^n=x^{m\times n}=x^{mn} [/tex]
In our case, x=a and the powers involved are m=8 and n=4.
Therefore, applying the power rule to the case given in our question we get:
[tex] (a^8)^4=a^{8\times 4}=a^{32} [/tex].
Therefore, Option D which says [tex] \boldsymbol{a^{32}} [/tex] is the correct option.
What is the factored form of x2 - 4x - 5 ?
solve the proportion
[tex] \frac{3}{4} = \frac{x}{6} [/tex]
A candy maker buys a bar of chocolate weighing 162 ounces. About how many pounds does the bar weigh?
Please explain how you worked the problem to find x!
1/x +1/(x+4) =1/5
Is the line through points P(0, 5) and Q(–1, 8) parallel to the line through points R(3, 3) and S(5, –1)? Explain.
a. No, the lines have unequal slopes.
b. Yes, the lines are both vertical.
c. Yes, the lines have equal slopes.
d. No, one line has slope, while the other has no slope.
$8000 invested at an APR of 6% for 10 years.
If interest is compounded annually, what is the amount of money after 10 years?
The formula in finding the maturity value is the following:
A = P (1 + r) ^t
Where A = Maturity value
P = principal amount
r = Annual percentage rate
t = time in years
Substituting the given amount to the formula:
A = $8,000 (1 + 6%) ^10
= $8,000 (1.7908)
= $14,326.40
Therefore, the amount of $8,000 after 10 years compounded annually at 6% is $14,326.40
PLEASE HELP!!!! How can you tell if a triangle can be drawn more than one way?
please explain in detail and be specific.
An equilateral triangle has a height of 26 cm. what is the length of each side of the triangle
Answer:
30.0 cm
Step-by-step explanation:
You want the side length of an equilateral triangle with a height of 26 cm.
Equilateral triangleThe altitude of the triangle divides it into two congruent 30°-60°-90° right triangles. The longer leg is the height: 26 cm. The hypotenuse is the longest of the three sides, which have the ratios ...
1 : √3 : 2
That is, the side of the equilateral triangle is 2/√3 times the altitude.
[tex]s = \dfrac{2}{\sqrt{3}}h = \dfrac{2(26\text{ cm})}{\sqrt{3}}\approx\boxed{30.0\text{ cm}}[/tex]
The length of each side is about 30.0 cm.
__
Additional comment
The side length is an irrational number of cm, approximately 30.022213997.... It rounds to 30.
Use the data to create a scatter plot. Year 0 1 2 3 4 Population 3 4 6 11 15 Use the point tool to plot the points from the table in the coordinate grid to create a scatter plot.
Answer:
Refer the attached figure.
Step-by-step explanation:
Given : Data of population
To find : Use the data to create a scatter plot?
Solution :
Data :
Year(x) Population(y)
0 3
1 4
2 6
3 11
4 15
A Scatter Plot has points that show the relationship between two sets of data.
With the help of scatter plot graph,
We plot the points of the data.
Refer the attached figure below.
Analyze the following budget, with an income of $750, to determine how much can be spent on food for the month. Month________ Budgeted Amount Food $___ Personal Items $20 Cell Phone $75 Entertainment $85 Car Expenses – Gas, Insurance $260 College Savings $250 a. No more than $70 can be spent on food. b. No more than $75 can be spent on food. c. No more than $80 can be spent on food. d. No more than $60 can be spent on food.
A monthly of $60 can be spent for food after all the expenses and savings
are sorted.
What is a numerical expression?A numerical expression is a mathematical statement written in the form of numbers and unknown variables. We can form numerical expressions from statements.
The numerical form of the given information can be formed as,
$(750 - 20 - 75 - 85 - 260 - 25).
= $60.
So, With an income of $750 and all the expenses including saving for for college worth of $250 the amount that can be spent on food is $60.
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Which phrase describes the variable expression m + 5?
Melissa had a balance of $104 in her checking account at the beginning of the month. She made one additional deposi of $216 and wrote checks $78 , $119 , and $105. What was the balance in her checking account at end of the month
Answer:
She only had $18 left
Step-by-step explanation:
How do I solve this equation for n? (6x2n)÷8=15
To solve the equation (6x2n)÷8=15 for n, you must perform algebraic operations to isolate n. This involves multiplying by 8, then dividing by 6x, and finally by 2, to arrive at n = 20 / x.
To solve the equation (6x2n)÷8=15 for n, we need to follow several steps. This involves manipulating the equation using algebraic operations until we isolate the variable n on one side. Here is the step-by-step process:
Multiply both sides of the equation by 8 to get rid of the fraction: 6x2n = 15 × 8.Simplify the right side: 6x2n = 120.Divide both sides by 6x to isolate n: 2n = 120 / 6x.Further simplify the right side to get n on its own: n = (120 / 6x) / 2.Complete the calculation based on the values of x that are given: n = 20 / x.If x is known, you can substitute the value of x into the equation to find the value of n. If x is not given, then this is the simplified form of the equation in terms of n and x.
In a circle with a radius of 36 3/5 cm, an arc is intercepted by a central angle of 2π7 radians.
What is the arc length?
Use 3.14 for π and round your final answer to the nearest hundredth.
Enter your answer as a decimal in the box.
cm
Answer:
The arc length is [tex]32.84\ cm[/tex]
Step-by-step explanation:
we know that
The circumference of a circle is equal to
[tex]C=2\pi r[/tex]
we have
[tex]r=36\frac{3}{5}\ cm=\frac{36*5+3}{5}=\frac{183}{5}\ cm[/tex]
substitute
[tex]C=2(3.14)(\frac{183}{5})=229.848\ cm[/tex]
Remember that
[tex]2\pi[/tex] radians subtends the complete circle of length [tex]229.848\ cm[/tex]
so
by proportion
Find the arc length by a central angle of [tex]2\pi/7[/tex] radians
[tex]\frac{229.848}{2\pi}=\frac{x}{2\pi/7}\\ \\x=229.848*(2\pi/7)/(2\pi)\\ \\x=`32.84\ cm[/tex]
The arc length is approximately 32.81 cm.
Step 1
To find the arc length intercepted by a central angle, we can use the formula:
[tex]\[ \text{Arc Length} = \text{radius} \times \text{central angle} \][/tex]
Given:
- Radius [tex](\( r \)[/tex]) = 36 3/5 cm
- Central angle [tex](\( \theta \)) = \( \frac{2\pi}{7} \)[/tex] radians
First, let's convert the radius to a decimal:
[tex]\[ \text{Radius} = 36 \frac{3}{5} \text{ cm} = 36.6 \text{ cm} \][/tex]
Step 2
Now, we can use the formula to find the arc length:
[tex]\[ \text{Arc Length} = 36.6 \times \frac{2\pi}{7} \]\[ \text{Arc Length} = 36.6 \times \frac{2 \times 3.14}{7} \]\[ \text{Arc Length} = 36.6 \times \frac{6.28}{7} \]\[ \text{Arc Length} = 36.6 \times 0.897 \]\[ \text{Arc Length} \approx 32.808 \][/tex]
Rounded to the nearest hundredth, the arc length is approximately 32.81 cm.
Which terms accurately classify this triangle? Choose exactly two answers that are correct. A. right B. acute C. isosceles D. scalene
A cube with a volume of 75cm^3 is dilated by a scale factor of 5. What is the volume of the dilated cube?
1) Which two lines are parallel?
I. 5y = -3x-5
II. 5y = -1 - 3x
III. 3y - 2x = -1
a. I. and II.
b. I. and III.
c. II. and III.
d. No two of the lines are parallel.
2) Which do lines are parallel?
I. 3y = -2x - 3
II. 3y = 1 - 2x
III. 4y - 3x = -1
a. I. and II.
c. II. and III.
b. I. and III.
d. No two of the lines are parallel.
A triangle has sides with lengths of 6 millimeters, 8 millimeters, and 10 millimeters. Is it a right triangle?
The sides 6 mm, 8 mm and 10 mm forms a right angles triangle.
We have,
The sides of the triangle are of lengths 6 mm, 8 mm and 10 mm.
The longest side length is 10 mm.
Now, the sum of squares of the smaller sides is given as:
6² + 8² = 36 + 64 = 100 = 10²
As, the sum of squares of the smaller sides is equal to the square of the longer side.
So, by the Pythagorean Theorem, the sides form a right triangle.
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wo groups of students were asked how many pets they had. The table below shows the numbers for each group: Group A 1 2 1 1 3 5 4 2 3 Group B 3 2 3 2 2 2 1 1 2 Based on the table, which of the following is true?
the interquartile range formula is the first quartile subtracted from the third quartile:
The question about how many pets two groups of students have cannot be answered without the correct data. Typically, such a question would involve computing descriptive statistics or organizing the data into a graph or table for easier comparison and interpretation.
Explanation:Based on the information given in the prompt, which seems to contain a mistake as it does not directly correlate to the question about the two groups of students and their number of pets, we can not effectively conclude which of the following is true regarding the data about Group A and Group B without the appropriate data or statements to compare. Therefore, we need the correct data to proceed with the analysis. Generally, in math problems like this, we would compute the mean, median, mode, or range of the data to compare the two sets, or we could use more advanced statistics if required.
In a correct scenario, you could group the data differently by organizing it into a frequency table or creating a graph to visualize the distribution. Depending on the data shape, you might group it by intervals or categories. Grouping data helps to understand and interpret the data more easily. For instance, it is often easier to see trends and patterns in a histogram or bar chart than in a raw list of numbers.
11. The polygons are similar, but not necessarily drawn to scale. Find the value of a and b.
5) If f(x) = 4x2 - 5x + 7 and g(x) = 3x2 - 2x + 8, find f(x) + g(x).