Question
Find the inverse of f(x)=5x3−1
algebraically.(1 point)
Responses
f−1(x)=3x+35
f inverse left parenthesis x right parenthesis equals Start Fraction 3 x plus 3 over 5 End Fraction
f−1(x)=3x5+1
f inverse left parenthesis x right parenthesis equals Start Fraction 3 x over 5 End Fraction plus 1
f−1(x)=5x+13
f inverse left parenthesis x right parenthesis equals Start Fraction 5 x plus 1 over 3 End Fraction
f−1(x)=3x+15
algebraically.(1 point)
Responses
f−1(x)=3x+35
f inverse left parenthesis x right parenthesis equals Start Fraction 3 x plus 3 over 5 End Fraction
f−1(x)=3x5+1
f inverse left parenthesis x right parenthesis equals Start Fraction 3 x over 5 End Fraction plus 1
f−1(x)=5x+13
f inverse left parenthesis x right parenthesis equals Start Fraction 5 x plus 1 over 3 End Fraction
f−1(x)=3x+15
Answers
Answered by
GPT-4o mini
To find the inverse of the function \( f(x) = 5x^3 - 1 \), we will follow these steps:
1. **Replace \( f(x) \) with \( y \)**:
\[
y = 5x^3 - 1
\]
2. **Solve for \( x \) in terms of \( y \)**:
First, add 1 to both sides:
\[
y + 1 = 5x^3
\]
Next, divide both sides by 5:
\[
\frac{y + 1}{5} = x^3
\]
Finally, take the cube root of both sides:
\[
x = \sqrt[3]{\frac{y + 1}{5}}
\]
3. **Replace \( x \) with \( f^{-1}(y) \) and \( y \) with \( x \)**:
Thus, the inverse function is:
\[
f^{-1}(x) = \sqrt[3]{\frac{x + 1}{5}}
\]
Now, let's rewrite this more clearly:
\[
f^{-1}(x) = \frac{\sqrt[3]{x + 1}}{5}
\]
Looking at the provided options, it seems none perfectly match. However, of those given, the correct form (if simplified to match structure) is:
\[
f^{-1}(x) = \frac{\sqrt[3]{x + 1}}{5}
\]
This does not visibly match any options provided.
If we check whether any options might approximate alternate transformations of the correct answer, none of the options listed (which seem to involve linear terms or simple fractions) align with the derived result. Therefore, the answer \( f^{-1}(x) = \sqrt[3]{\frac{x + 1}{5}} \) is the complete answer which matches with our calculations, but it is not in the provided options.
1. **Replace \( f(x) \) with \( y \)**:
\[
y = 5x^3 - 1
\]
2. **Solve for \( x \) in terms of \( y \)**:
First, add 1 to both sides:
\[
y + 1 = 5x^3
\]
Next, divide both sides by 5:
\[
\frac{y + 1}{5} = x^3
\]
Finally, take the cube root of both sides:
\[
x = \sqrt[3]{\frac{y + 1}{5}}
\]
3. **Replace \( x \) with \( f^{-1}(y) \) and \( y \) with \( x \)**:
Thus, the inverse function is:
\[
f^{-1}(x) = \sqrt[3]{\frac{x + 1}{5}}
\]
Now, let's rewrite this more clearly:
\[
f^{-1}(x) = \frac{\sqrt[3]{x + 1}}{5}
\]
Looking at the provided options, it seems none perfectly match. However, of those given, the correct form (if simplified to match structure) is:
\[
f^{-1}(x) = \frac{\sqrt[3]{x + 1}}{5}
\]
This does not visibly match any options provided.
If we check whether any options might approximate alternate transformations of the correct answer, none of the options listed (which seem to involve linear terms or simple fractions) align with the derived result. Therefore, the answer \( f^{-1}(x) = \sqrt[3]{\frac{x + 1}{5}} \) is the complete answer which matches with our calculations, but it is not in the provided options.