Question

find the inverse of $f(x)=-\frac{1}{2}x + 4$. the inverse is $g(x)=square$. graph the function and its inverse.

Explanation:

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

$y = -\frac{1}{2}x + 4$

Step2: Swap $x$ and $y$

$x = -\frac{1}{2}y + 4$

Step3: Isolate the term with $y$

$x - 4 = -\frac{1}{2}y$

Step4: Solve for $y$

$y = -2(x - 4) = -2x + 8$

Step5: Rename $y$ as $g(x)$

$g(x) = -2x + 8$

For graphing:

• The original function $f(x) = -\frac{1}{2}x + 4$ has a y-intercept at $(0,4)$ and x-intercept at $(8,0)$ (already plotted).
• The inverse function $g(x) = -2x + 8$ has a y-intercept at $(0,8)$ and x-intercept at $(4,0)$. Plot these two points and draw a straight line through them; this line is the reflection of $f(x)$ over the line $y=x$.

Answer:

$g(x) = -2x + 8$
(For graphing: Plot the line through points $(0,8)$ and $(4,0)$ as the inverse function)