To find the inverse function [tex]\( f^{-1}(x) \)[/tex] for [tex]\( f(x) = \sqrt[3]{7x + 6} \)[/tex], follow these steps:
1. Express [tex]\( y = f(x) \)[/tex]:
[tex]\[
y = \sqrt[3]{7x + 6}
\][/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[
y = (7x + 6)^{1/3}
\][/tex]
3. Cube both sides of the equation to eliminate the cube root:
[tex]\[
y^3 = 7x + 6
\][/tex]
4. Isolate [tex]\( x \)[/tex] on one side of the equation:
[tex]\[
y^3 - 6 = 7x
\][/tex]
5. Divide both sides by 7 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{y^3 - 6}{7}
\][/tex]
6. Express the inverse function, [tex]\( f^{-1}(x) \)[/tex], using the simplified form:
[tex]\[
f^{-1}(x) = \frac{x^3 - 6}{7}
\][/tex]
7. Simplify the fraction:
[tex]\[
f^{-1}(x) = \frac{1}{7} x^3 - \frac{6}{7}
\][/tex]
So, the inverse function is:
[tex]\[
f^{-1}(x) = 0.142857142857143x^3 - 0.857142857142857
\][/tex]
This is the detailed step-by-step process for finding the inverse function [tex]\( f^{-1}(x) \)[/tex].
