To find the inverse of the function [tex]\( h(x) = \frac{3}{4}x + 12 \)[/tex], we need to follow these steps:
1. Replace [tex]\( h(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = \frac{3}{4}x + 12 \][/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
- Start by isolating the term containing [tex]\( x \)[/tex]:
[tex]\[ y - 12 = \frac{3}{4}x \][/tex]
- Next, to solve for [tex]\( x \)[/tex], 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:
We typically swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to express the inverse function correctly:
[tex]\[ h^{-1}(x) = \frac{4}{3}(x - 12) \][/tex]
Therefore, the inverse of the function [tex]\( h(x) = \frac{3}{4}x + 12 \)[/tex] is:
[tex]\[ h^{-1}(x) = \frac{4}{3}(x - 12) \][/tex]
