Question

directions: read each problem and answer carefully. explain your work.
show mathematical steps

1. graph ( f(x) = (x - 1)^3 - 2 ) and its inverse on the same set of axes.
2. write the equation of the inverse for each of the following functions.

a) ( f(x) = 3x - 8 )
b) ( f(x) = \frac{1}{2}x + 6 )
c) ( f(x) = \frac{x + 6}{2} )

Explanation:

Problem 1: Graph $f(x)=(x-1)^3 - 2$ and its inverse

Step1: Identify parent function shifts

Parent function: $y=x^3$. Shift right 1, down 2: $f(x)=(x-1)^3 - 2$

Step2: Find key points for $f(x)$

For $x=-1$: $f(-1)=(-2)^3-2=-10$; $x=1$: $f(1)=0-2=-2$; $x=2$: $f(2)=1-2=-1$; $x=0$: $f(0)=(-1)^3-2=-3$

Step3: Find inverse function

Swap $x$ and $y$: $x=(y-1)^3 - 2$. Solve for $y$:
$x+2=(y-1)^3$ → $y=\sqrt[3]{x+2}+1$

Step4: Find key points for inverse

Swap $x,y$ from $f(x)$ points: $(-10,-1)$, $(-2,1)$, $(-1,2)$, $(-3,0)$

Step5: Plot and reflect

Plot $f(x)$ through its key points, then plot inverse points (reflected over $y=x$).

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Problem 2: Find inverse equations

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Part a) $f(x)=3x-8$

Step1: Replace $f(x)$ with $y$

$y=3x-8$

Step2: Swap $x$ and $y$

$x=3y-8$

Step3: Solve for $y$

$x+8=3y$ → $y=\frac{x+8}{3}$

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Part b) $f(x)=\frac{1}{2}x+6$

Step1: Replace $f(x)$ with $y$

$y=\frac{1}{2}x+6$

Step2: Swap $x$ and $y$

$x=\frac{1}{2}y+6$

Step3: Solve for $y$

$x-6=\frac{1}{2}y$ → $y=2(x-6)=2x-12$

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Part c) $f(x)=\frac{x+6}{2}$

Step1: Replace $f(x)$ with $y$

$y=\frac{x+6}{2}$

Step2: Swap $x$ and $y$

$x=\frac{y+6}{2}$

Step3: Solve for $y$

$2x=y+6$ → $y=2x-6$

Answer:

1. (Graph: $f(x)=(x-1)^3-2$ plotted with key points $(-1,-10), (0,-3), (1,-2), (2,-1)$; inverse $y=\sqrt[3]{x+2}+1$ plotted with reflected points $(-10,-1), (-3,0), (-2,1), (-1,2)$, both reflected over $y=x$)
2. a) $f^{-1}(x)=\frac{x+8}{3}$

b) $f^{-1}(x)=2x-12$
c) $f^{-1}(x)=2x-6$