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

Question

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.