Use the table to answer the question.

Question

x    	y 
0	 −5 
1	 −9 
2	 −13 
3	 −17 
Which of the following equations correctly describes the function in the table using function notation?
(1 point)
Responses

f(x)=−5x−4

f of x is equal to negative 5 x minus 4

f(y)=−5x−4
f left parenthesis y right parenthesis equals negative 5 x minus 4

f(x)=−4x−5
f left parenthesis x right parenthesis equals negative 4 x minus 5

f(y)=−4x−5

GPT-4o mini · Answer

To determine the correct function that describes the relationship in the table, let's analyze the values provided.

The table shows the following pairs of $ (x, y) $:

For $ x = 0, y = -5 $
For $ x = 1, y = -9 $
For $ x = 2, y = -13 $
For $ x = 3, y = -17 $

By examining the changes in $ y $ as $ x $ increases, we can identify the pattern.

The changes in $ y $:

From $ -5 $ to $ -9 $ is a decrease of 4 ($-9 - (-5) = -4$)
From $ -9 $ to $ -13 $ is a decrease of 4 ($-13 - (-9) = -4$)
From $ -13 $ to $ -17 $ is a decrease of 4 ($-17 - (-13) = -4$)

This indicates that the function is linear with a slope of -4.

To find the equation of the line, we can use the slope-intercept form $ y = mx + b $, where $ m $ is the slope and $ b $ is the y-intercept. Given the slope is -4, we can start with:

\[ y = -4x + b \]

Now, we need to find $ b $. We can use any of the points; for example, let's use $ (0, -5) $:

\[ -5 = -4(0) + b \] \[ b = -5 \]

Thus, the equation becomes:

\[ y = -4x - 5 \]

In function notation, we write this as:

\[ f(x) = -4x - 5 \]

Therefore, the correct response is:

f(x)=−4x−5

Answer

x y
0 -8
1 -7 1/2
2 -7
3 -6 1/2
Which equation describes the function shown in the mapping diagram?

(1 point)
Responses

f(y)=8x−12
f left parenthesis y right parenthesis equals 8 x minus Start Fraction 1 over 2 End Fraction

f(y)=12x−8
f left parenthesis y right parenthesis equals Start Fraction 1 over 2 End Fraction x minus 8

f(x)=8x−12
f left parenthesis x right parenthesis equals 8 x minus Start Fraction 1 over 2 End Fraction

f(x)=12x−8

GPT-4o mini · Answer

To determine the correct function that describes the relationship in the provided table, we first analyze the data points:

For $ x = 0, y = -8 $
For $ x = 1, y = -7.5 $
For $ x = 2, y = -7 $
For $ x = 3, y = -6.5 $

Next, let's identify how $ y $ changes as $ x $ increases.

The changes in $ y $ based on the increments of $ x $ are:

From $ -8 $ to $ -7.5 $ is an increase of $ 0.5 $ (i.e., $-7.5 - (-8) = 0.5$)
From $ -7.5 $ to $ -7 $ is an increase of $ 0.5 $
From $ -7 $ to $ -6.5 $ is an increase of $ 0.5 $

Since every time $ x $ increases by 1, $ y $ increases by 0.5, we can conclude that the slope $ m $ of the linear relationship is $ 0.5 $.

We can express the linear function in slope-intercept form $ y = mx + b $, where $ m $ is the slope and $ b $ is the y-intercept. For our case, we have:

\[ y = 0.5x + b \]

Now, we need to find $ b $. We can use one of the points; let's use $ (0, -8) $:

\[ -8 = 0.5(0) + b \] \[ b = -8 \]

So the equation becomes:

\[ y = 0.5x - 8 \]

In function notation, we express this as:

\[ f(x) = 0.5x - 8 \]

We should express $ 0.5 $ as a fraction: $ \frac{1}{2} $, so we get:

\[ f(x) = \frac{1}{2}x - 8 \]

Now, let's match this with the provided responses:

f(y)=8x−12
f(y)=12x−8
f(x)=8x−12
f(x)=12x−8

None of these directly matches $ f(x) = \frac{1}{2}x - 8 $.

However, based on the slope of $ 0.5 $ or $ \frac{1}{2} $, the closest relevant options seem to be limited to errors in the formulation of options.

I'll clarify that the correct way to represent the function from our analysis is actually not listed, but none of the provided options accurately represent the calculated function.

Please double-check the options, as the derived function is:

f(x) = \frac{1}{2}x - 8

Answer

Evaluate the function for the following input: f(x)=3−2x . Which input generates the largest output when evaluating the function?(1 point)
Responses

f(3)
f left parenthesis 3 right parenthesis

f(5)
f left parenthesis 5 right parenthesis

f(2)
f left parenthesis 2 right parenthesis

f(0)

GPT-4o mini · Answer

To determine which input generates the largest output for the function $ f(x) = 3 - 2x $, we will evaluate the function for each of the given inputs: $ f(0) $, $ f(2) $, $ f(3) $, and $ f(5) $.

Evaluate $ f(0) $: \[ f(0) = 3 - 2(0) = 3 - 0 = 3 \]
Evaluate $ f(2) $: \[ f(2) = 3 - 2(2) = 3 - 4 = -1 \]
Evaluate $ f(3) $: \[ f(3) = 3 - 2(3) = 3 - 6 = -3 \]
Evaluate $ f(5) $: \[ f(5) = 3 - 2(5) = 3 - 10 = -7 \]

Now we summarize the outputs:

$ f(0) = 3 $
$ f(2) = -1 $
$ f(3) = -3 $
$ f(5) = -7 $

The largest output is 3, which corresponds to the input $ f(0) $.

Thus, the input that generates the largest output is:

f(0).

Answer

The function V(r)=10πr2 represents the volume of a cylinder that is 10 inches high with a radius of r. Which is the correct interpretation for the function notation V(3)=90π ?(1 point)
Responses

The volume of the cylinder is 3 when the radius is 90π.
The volume of the cylinder is 3 when the radius is 90 pi .

A radius of 3 gives the value 90.
A radius of 3 gives the value 90.

The volume of the cylinder is 90π when the radius is 3.
The volume of the cylinder is 90 pi when the radius is 3.

A cylinder of 90π is produced from a radius of 3.

GPT-4o mini · Answer

To interpret the function notation $ V(3) = 90\pi $ in the context of the given function $ V(r) = 10\pi r^2 $, we need to analyze what this means. 1. The function $ V(r) $ represents the volume of a cylinder with a height of 10 inches and radius $ r $. 2. The expression $ V(3) = 90\pi $ indicates that when the radius $ r $ is 3 inches, the volume $ V $ of the cylinder is equal to $ 90\pi $ cubic inches. Let's examine your provided responses: 1. **The volume of the cylinder is 3 when the radius is 90π.** 2. **The volume of the cylinder is 3 when the radius is 90 pi.** 3. **A radius of 3 gives the value 90.** 4. **A radius of 3 gives the value 90.** 5. **The volume of the cylinder is 90π when the radius is 3.** 6. **The volume of the cylinder is 90 pi when the radius is 3.** 7. **A cylinder of 90π is produced from a radius of 3.** The correct interpretation for $ V(3) = 90\pi $ is captured by statements 5 and 6: - **The volume of the cylinder is 90π when the radius is 3.** - **The volume of the cylinder is 90 pi when the radius is 3.** Either statement 5 or 6 accurately describes the relationship indicated by the function notation $ V(3) = 90\pi $. Therefore, the correct answer is: **The volume of the cylinder is 90π when the radius is 3.**

Answer

Monique is punting a football and the height the ball travels is a function of time. The function h(s)=−s2+10s+3 gives the height of the ball, h(s) , given the number of seconds, s , that the ball has been in the air. After flying through the air for 7.5 seconds, what is the height of the ball? Find h(7.5) .(1 point)
Responses

21.75
21.75

134.25
134.25

21.5
21.5

24

GPT-4o mini · Answer

To find the height of the ball after 7.5 seconds, we need to evaluate the function $ h(s) = -s^2 + 10s + 3 $ at $ s = 7.5 $.

Let's calculate $ h(7.5) $:

\[ h(7.5) = -(7.5)^2 + 10(7.5) + 3 \]

Calculating each term separately:

$ (7.5)^2 = 56.25 $
$ 10(7.5) = 75 $

Now we can substitute these values back into the function:

\[ h(7.5) = -56.25 + 75 + 3 \]

Now perform the additions and subtractions:

\[ h(7.5) = -56.25 + 75 + 3 = -56.25 + 78 = 21.75 \]

Therefore, the height of the ball after 7.5 seconds is:

21.75