Solved on Dec 08, 2023

Determine if $t(x)$ is a linear function given the values in the table: $x = \{0, 1, 2, 3\}$ , $f(x) = \{-8, -4, 0, 4\}$ . Select: A. $f(x) =$ __ or B. The function is not linear.

STEP 1

Assumptions
1. The function $t(x)$ is given by the table with $x$ values 0, 1, 2, and 3.
2. The corresponding $f(x)$ values are -8, -4, 0, and 4 respectively.
3. A linear function has the form $f(x) = mx + b$ where $m$ is the slope and $b$ is the y-intercept.
4. To determine if $t(x)$ is linear, we must check if the differences between consecutive $f(x)$ values are constant.

STEP 2

First, we need to find the differences between consecutive $f(x)$ values to determine if they are constant, which would indicate a constant slope.
$\Delta f(x)_{1} = f(x_{1}) - f(x_{0})$ $\Delta f(x)_{2} = f(x_{2}) - f(x_{1})$ $\Delta f(x)_{3} = f(x_{3}) - f(x_{2})$

STEP 3

Now, plug in the given values for $f(x)$ to calculate the differences.
$\Delta f(x)_{1} = f(1) - f(0) = -4 - (-8)$ $\Delta f(x)_{2} = f(2) - f(1) = 0 - (-4)$ $\Delta f(x)_{3} = f(3) - f(2) = 4 - 0$

STEP 4

Calculate the differences.
$\Delta f(x)_{1} = -4 - (-8) = -4 + 8 = 4$ $\Delta f(x)_{2} = 0 - (-4) = 0 + 4 = 4$ $\Delta f(x)_{3} = 4 - 0 = 4$

STEP 5

Since all the differences $\Delta f(x)_{1}$ , $\Delta f(x)_{2}$ , and $\Delta f(x)_{3}$ are equal to 4, the slope $m$ is constant.
$m = 4$

STEP 6

To find the y-intercept $b$ , we can use the value of $f(x)$ when $x = 0$ .
$b = f(0) = -8$

STEP 7

Now we can write the equation of the linear function $t(x)$ using the slope $m$ and y-intercept $b$ .
$f(x) = mx + b$

STEP 8

Plug in the values for $m$ and $b$ to get the linear function.
$f(x) = 4x - 8$

STEP 9

Since we have determined that the differences between consecutive $f(x)$ values are constant and we have found a linear function that fits the given points, the values could represent the linear function $f(x) = 4x - 8$ .
The correct choice is A, and the linear function is $f(x) = 4x - 8$ .

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Determine if $t(x)$ is a linear function given the values in the table: $x = \{0, 1, 2, 3\}$ , $f(x) = \{-8, -4, 0, 4\}$ . Select: A. $f(x) =$ __ or B. The function is not linear.

STEP 1

STEP 2

First, we need to find the differences between consecutive $f(x)$ values to determine if they are constant, which would indicate a constant slope.
$\Delta f(x)_{1} = f(x_{1}) - f(x_{0})$ $\Delta f(x)_{2} = f(x_{2}) - f(x_{1})$ $\Delta f(x)_{3} = f(x_{3}) - f(x_{2})$

STEP 3

Now, plug in the given values for $f(x)$ to calculate the differences.
$\Delta f(x)_{1} = f(1) - f(0) = -4 - (-8)$ $\Delta f(x)_{2} = f(2) - f(1) = 0 - (-4)$ $\Delta f(x)_{3} = f(3) - f(2) = 4 - 0$

STEP 4

Calculate the differences.
$\Delta f(x)_{1} = -4 - (-8) = -4 + 8 = 4$ $\Delta f(x)_{2} = 0 - (-4) = 0 + 4 = 4$ $\Delta f(x)_{3} = 4 - 0 = 4$

STEP 5

Since all the differences $\Delta f(x)_{1}$ , $\Delta f(x)_{2}$ , and $\Delta f(x)_{3}$ are equal to 4, the slope $m$ is constant.
$m = 4$

STEP 6

To find the y-intercept $b$ , we can use the value of $f(x)$ when $x = 0$ .
$b = f(0) = -8$

STEP 7

Now we can write the equation of the linear function $t(x)$ using the slope $m$ and y-intercept $b$ .
$f(x) = mx + b$

STEP 8

Plug in the values for $m$ and $b$ to get the linear function.
$f(x) = 4x - 8$

STEP 9

Since we have determined that the differences between consecutive $f(x)$ values are constant and we have found a linear function that fits the given points, the values could represent the linear function $f(x) = 4x - 8$ .
The correct choice is A, and the linear function is $f(x) = 4x - 8$ .

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Determine if t(x)t(x)t(x) is a linear function given the values in the table: x={0,1,2,3}x = \{0, 1, 2, 3\}x={0,1,2,3}, f(x)={−8,−4,0,4}f(x) = \{-8, -4, 0, 4\}f(x)={−8,−4,0,4}. Select: A. f(x)=f(x) = f(x)=__ or B. The function is not linear.

STEP 1

STEP 2

STEP 3

STEP 4

Calculate the differences.Δf(x)1=−4−(−8)=−4+8=4\Delta f(x)_{1} = -4 - (-8) = -4 + 8 = 4Δf(x)1​=−4−(−8)=−4+8=4 Δf(x)2=0−(−4)=0+4=4\Delta f(x)_{2} = 0 - (-4) = 0 + 4 = 4Δf(x)2​=0−(−4)=0+4=4 Δf(x)3=4−0=4\Delta f(x)_{3} = 4 - 0 = 4Δf(x)3​=4−0=4

STEP 5

Since all the differences Δf(x)1\Delta f(x)_{1}Δf(x)1​, Δf(x)2\Delta f(x)_{2}Δf(x)2​, and Δf(x)3\Delta f(x)_{3}Δf(x)3​ are equal to 4, the slope mmm is constant.m=4m = 4m=4

STEP 6

To find the y-intercept bbb, we can use the value of f(x)f(x)f(x) when x=0x = 0x=0.b=f(0)=−8b = f(0) = -8b=f(0)=−8

STEP 7

Now we can write the equation of the linear function t(x)t(x)t(x) using the slope mmm and y-intercept bbb.f(x)=mx+bf(x) = mx + bf(x)=mx+b

STEP 8

Plug in the values for mmm and bbb to get the linear function.f(x)=4x−8f(x) = 4x - 8f(x)=4x−8

STEP 9

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Determine if $t(x)$ is a linear function given the values in the table: $x = \{0, 1, 2, 3\}$ , $f(x) = \{-8, -4, 0, 4\}$ . Select: A. $f(x) =$ __ or B. The function is not linear.

Calculate the differences.
$\Delta f(x)_{1} = -4 - (-8) = -4 + 8 = 4$ $\Delta f(x)_{2} = 0 - (-4) = 0 + 4 = 4$ $\Delta f(x)_{3} = 4 - 0 = 4$

Since all the differences $\Delta f(x)_{1}$ , $\Delta f(x)_{2}$ , and $\Delta f(x)_{3}$ are equal to 4, the slope $m$ is constant.
$m = 4$

To find the y-intercept $b$ , we can use the value of $f(x)$ when $x = 0$ .
$b = f(0) = -8$

Now we can write the equation of the linear function $t(x)$ using the slope $m$ and y-intercept $b$ .
$f(x) = mx + b$

Plug in the values for $m$ and $b$ to get the linear function.
$f(x) = 4x - 8$