Question

For the function f, of, x, equals, left parenthesis, x, minus, 10, right parenthesis, start superscript, one third, end superscriptf(x)=(x−10)
3
1
​

, find f, to the power minus 1 , left parenthesis, x, right parenthesisf
−1
(x).
Answer
Attempt 1 out of 2
Multiple Choice Answers
f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, left parenthesis, x, minus, 10, right parenthesis, cubedf
−1
(x)=(x−10)
3

f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, x, cubed, plus, 10f
−1
(x)=x
3
+10
f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, x, start superscript, one third, end superscript, plus, 10f
−1
(x)=x
3
1
​

+10
f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, left parenthesis, x, plus, 10, right parenthesis, cubedf
−1
(x)=(x+10)
3

Answer 1

To find the inverse function \( f^{-1}(x) \) for the function defined as

\[
f(x) = (x - 10)^{1/3},
\]

we start by letting \( y = f(x) \):

\[
y = (x - 10)^{1/3}.
\]

Next, we will solve for \( x \) in terms of \( y \).

1. **Cube both sides:**
\[
y^3 = x - 10.
\]

2. **Isolate \( x \):**
\[
x = y^3 + 10.
\]

Now we have expressed \( x \) in terms of \( y \). To find the inverse function, we can replace \( y \) with \( x \):

\[
f^{-1}(x) = x^3 + 10.
\]

So, the correct answer is:

\[
f^{-1}(x) = x^3 + 10.
\]

Thus, the answer option that matches is:

**f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, x, cubed, plus, 10.**