Question

Question
What is the inverse of f(x)=2x3−7
?(1 point)
Responses

f−1(x)=x−72−−−√3
f inverse left parenthesis x right parenthesis equals root index 3 Start Root Start Fraction x minus 7 over 2 End Fraction End Root

f−1(x)=x+72−−−√3
f inverse left parenthesis x right parenthesis equals root index 3 Start Root Start Fraction x plus 7 over 2 End Fraction End Root

f−1(x)=2x−7−−−−−√3
f inverse left parenthesis x right parenthesis equals root index 3 Start Root 2 x minus 7 End Root

f−1(x)=2x+7−−−−−√3

Answer 1

To find the inverse of the function \( f(x) = 2x^3 - 7 \), we need to follow these steps:

1. Replace \( f(x) \) with \( y \):
\[
y = 2x^3 - 7
\]

2. Solve for \( x \) in terms of \( y \):
\[
y + 7 = 2x^3
\]
\[
\frac{y + 7}{2} = x^3
\]
\[
x = \sqrt[3]{\frac{y + 7}{2}}
\]

3. Replace \( y \) with \( x \) to express the inverse function:
\[
f^{-1}(x) = \sqrt[3]{\frac{x + 7}{2}}
\]

So, the correct response is:
\[
f^{-1}(x) = \sqrt[3]{\frac{x + 7}{2}}
\]

This corresponds to the second option in your list:
**f−1(x) = root index 3 Start Root Start Fraction x + 7 over 2 End Fraction End Root**.