Question

1. complete the following.

part a. find the inverse of $f(x) = 6x - 3$.

$\boldsymbol{f^{-1}(x) = \square}$

part b. graph the function and its inverse.

Explanation:

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

$y = 6x - 3$

Step2: Swap $x$ and $y$

$x = 6y - 3$

Step3: Solve for $y$

Add 3 to both sides: $x + 3 = 6y$
Divide by 6: $y = \frac{x + 3}{6} = \frac{1}{6}x + \frac{1}{2}$

Step4: Rename $y$ as $f^{-1}(x)$

$f^{-1}(x) = \frac{x + 3}{6}$

Answer:

Part A:

$f^{-1}(x) = \frac{x + 3}{6}$

Part B:

1. For $f(x) = 6x - 3$: This is a linear line with slope $6$ and y-intercept $(0, -3)$. Plot the y-intercept, then use the slope (rise 6, run 1) to plot a second point and draw the line.
2. For $f^{-1}(x) = \frac{1}{6}x + \frac{1}{2}$: This is a linear line with slope $\frac{1}{6}$ and y-intercept $(0, \frac{1}{2})$. Plot the y-intercept, then use the slope (rise 1, run 6) to plot a second point and draw the line.
3. Additionally, the graph of a function and its inverse are reflections over the line $y=x$, so you can also plot $f(x)$, reflect its points over $y=x$, and connect them to get $f^{-1}(x)$.