Question

find the inverse of \\( f(x) = 6x \\).\
the inverse is \\( g(x) = \square \\).\
graph the function and its inverse.

Explanation:

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

$y = 6x$

Step2: Swap $x$ and $y$

$x = 6y$

Step3: Solve for $y$

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

Step4: Rename $y$ as $g(x)$

$g(x) = \frac{x}{6}$

For graphing:

• The original function $f(x)=6x$ is a line with slope $6$ passing through the origin $(0,0)$. Plot points like $(0,0)$ and $(1,6)$ then draw a line through them.
• The inverse function $g(x)=\frac{x}{6}$ is a line with slope $\frac{1}{6}$ passing through the origin $(0,0)$. Plot points like $(0,0)$ and $(6,1)$ then draw a line through them. The two lines are reflections over the line $y=x$.

Answer:

$g(x) = \frac{x}{6}$

Graph details:

1. For $f(x)=6x$: Draw a line through $(0,0)$ and $(1,6)$.
2. For $g(x)=\frac{x}{6}$: Draw a line through $(0,0)$ and $(6,1)$.