To find the inverse of the given function [tex]\(h(x) = \frac{2x - 4}{3}\)[/tex], we need to follow these steps:
1. Express the function in terms of [tex]\(y\)[/tex]:
[tex]\[
y = \frac{2x - 4}{3}
\][/tex]
2. Swap [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
To find the inverse function, we switch the roles of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] in the equation:
[tex]\[
x = \frac{2y - 4}{3}
\][/tex]
3. Solve for [tex]\(y\)[/tex]:
We now solve this equation for [tex]\(y\)[/tex]:
[tex]\[
x = \frac{2y - 4}{3}
\][/tex]
To isolate [tex]\(y\)[/tex], we first multiply both sides by 3:
[tex]\[
3x = 2y - 4
\][/tex]
4. Isolate [tex]\(y\)[/tex]:
Add 4 to both sides of the equation:
[tex]\[
3x + 4 = 2y
\][/tex]
5. Divide by 2:
Finally, divide by 2 to isolate [tex]\(y\)[/tex]:
[tex]\[
y = \frac{3x + 4}{2}
\][/tex]
Thus, the inverse function [tex]\(h^{-1}(x)\)[/tex] is:
[tex]\[
h^{-1}(x) = \frac{3x + 4}{2}
\][/tex]
This is the inverse of the given function [tex]\(h(x) = \frac{2x - 4}{3}\)[/tex].
