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

Question

5
x
3
1
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+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: \[ \boxed{f^{-1}(x) = 125(x - 7)^3} \]