To find the inverse [tex]\( f^{-1}(x) \)[/tex] of the function [tex]\( f(x) = \sqrt[3]{x-7} + 9 \)[/tex], we will follow these steps:
1. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[
y = \sqrt[3]{x-7} + 9
\][/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
- First, isolate the cube root term by subtracting 9 from both sides:
[tex]\[
y - 9 = \sqrt[3]{x-7}
\][/tex]
- Next, cube both sides to remove the cube root:
[tex]\[
(y - 9)^3 = x - 7
\][/tex]
- Finally, solve for [tex]\( x \)[/tex] by adding 7 to both sides:
[tex]\[
x = (y - 9)^3 + 7
\][/tex]
3. Express the inverse function:
- Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] to get the inverse function:
[tex]\[
f^{-1}(x) = (x - 9)^3 + 7
\][/tex]
Thus, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[
f^{-1}(x) = (x - 9)^3 + 7
\][/tex]
