Question

For the function f, of, x, equals, start fraction, x, start superscript, one third, end superscript, divided by, 5, end fraction, plus, 7f(x)=
5
x
3
1
​


​
+7, find f, to the power minus 1 , left parenthesis, x, right parenthesisf
−1
(x)

Answer 1

To find the inverse function \( f^{-1}(x) \) for the given function \( f(x) = \frac{x^{\frac{1}{3}}}{5} + 7 \), we need to follow a systematic approach.

1. **Set \( f(x) \) equal to \( y \)**:
\[
y = \frac{x^{\frac{1}{3}}}{5} + 7
\]

2. **Solve for \( x \) in terms of \( y \)**:
Start by isolating the term with \( x \):
\[
y - 7 = \frac{x^{\frac{1}{3}}}{5}
\]
Multiply both sides by 5:
\[
5(y - 7) = x^{\frac{1}{3}}
\]

3. **Cube both sides to eliminate the cube root**:
\[
(5(y - 7))^3 = x
\]

4. **Express \( x \) in terms of \( y \)**:
\[
x = 125(y - 7)^3
\]

5. **Interchanging \( x \) and \( y \) to find the inverse**:
We swap \( x \) and \( y \):
\[
y = 125(x - 7)^3
\]

Thus, the inverse function is:
\[
f^{-1}(x) = 125(x - 7)^3
\]

So the final answer for the inverse function is:
\[
\boxed{f^{-1}(x) = 125(x - 7)^3}
\]