Question

features of functions
name:
date:
period:

1. find the equation of $f^{-1}(x)$.

a. $f(x)=3x + 6$

1. the graph of $g(x)$ is given.

graph $g^{-1}(x)$ (use at least 3 points).
piecewise & absolute value

1. given the absolute value function below, write it in piecewise form. when writing in piecewise form, be sure to include the intervals for each sub function. then graph the function.

absolute value function: $f(x) = |x - 4|+2$
piecewise form:
$f(x)=\

$$\begin{cases} \\\\ \\\\ \\end{cases}$$

$

Explanation:

Problem 1

Step1: Set $y=f(x)$

$y = 3x + 6$

Step2: Swap $x$ and $y$

$x = 3y + 6$

Step3: Solve for $y$

$x - 6 = 3y$
$\frac{x - 6}{3} = y$
$y = \frac{1}{3}x - 2$

Step1: Identify 3 points on $g(x)$

Choose points like $(0, 6)$, $(2, 0)$, $(4, -2)$ from the graph.

Step2: Swap $x,y$ for inverse points

Inverse points: $(6, 0)$, $(0, 2)$, $(-2, 4)$

Step3: Plot and connect points

Plot the 3 inverse points and draw a smooth curve through them to graph $g^{-1}(x)$.

Step1: Find vertex of absolute value

The vertex of $f(x)=|x-4|+2$ is at $x=4$.

Step2: Define piecewise intervals

For $x \geq 4$, $|x-4|=x-4$; for $x < 4$, $|x-4|=-(x-4)$.

Step3: Write sub-functions

For $x \geq 4$: $f(x) = (x-4) + 2 = x - 2$
For $x < 4$: $f(x) = -(x-4) + 2 = -x + 6$

Step4: Graph the function

Plot the vertex at $(4,2)$, draw the line $y=x-2$ for $x \geq 4$ and $y=-x+6$ for $x < 4$.

Answer:

$f^{-1}(x) = \frac{1}{3}x - 2$

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Problem 2