For the function f, of, x, equals, start fraction, x, cubed, plus, 8, divided by, 9, end fractionf(x)=
9
x
3
+8
, find f, to the power minus 1 , left parenthesis, x, right parenthesisf
−1
(x).
Answer
Multiple Choice Answers
f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, cube root of, 9, x, minus, 8, end cube rootf
−1
(x)=
3
9x−8
f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, cube root of, 9, x, end cube root, minus, 8f
−1
(x)=
3
9x
−8
f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, 9, cube root of, x, minus, 8, end cube rootf
−1
(x)=9
3
x−8
f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, 9, left parenthesis, cube root of, x, end cube root, minus, 8, right parenthesisf
−1
(x)=9(
3
x
−8)
1 answer
To find the inverse function \( f^{-1}(x) \) of the given function \( f(x) = \frac{x^3 + 8}{9} \), we will follow the steps to isolate \( x \) in terms of \( y \).
Start with the equation:
\[
y = \frac{x^3 + 8}{9}
\]
Multiply both sides by 9 to eliminate the fraction:
\[
9y = x^3 + 8
\]
Subtract 8 from both sides:
\[
9y - 8 = x^3
\]
Take the cube root of both sides to solve for \( x \):
\[
x = \sqrt[3]{9y - 8}
\]
To express the inverse function, switch \( x \) and \( y \):
\[
f^{-1}(x) = \sqrt[3]{9x - 8}
\]