To find the inverse of the function [tex]\( h(x) = \frac{3}{4} x + 12 \)[/tex], we follow these steps:
1. Rewrite the function using [tex]\( y \)[/tex] to denote the output:
[tex]\[
y = \frac{3}{4} x + 12
\][/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
- First, we isolate [tex]\( x \)[/tex]. Start by subtracting 12 from both sides of the equation:
[tex]\[
y - 12 = \frac{3}{4} x
\][/tex]
- Next, to solve for [tex]\( x \)[/tex], we need to eliminate the fraction. Multiply both sides of the equation by the reciprocal of [tex]\( \frac{3}{4} \)[/tex], which is [tex]\( \frac{4}{3} \)[/tex]:
[tex]\[
x = \frac{4}{3} (y - 12)
\][/tex]
3. Express the inverse function [tex]\( h^{-1}(x) \)[/tex]:
- We replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] to represent the input variable of the inverse function:
[tex]\[
h^{-1}(x) = \frac{4}{3} (x - 12)
\][/tex]
Thus, the inverse of the function [tex]\( h(x) \)[/tex] is:
[tex]\[
h^{-1}(x) = \frac{4}{3} (x - 12)
\][/tex]
