Question

For the function f(x)=(x−4)13
, find f−1(x)
.(1 point)
Responses

f−1(x)=x3+4
f inverse of x is equal to x cubed plus 4

f−1(x)=(x−4)3
f inverse of x is equal to open paren x minus 4 close paren cubed

x13+4
x raised to the 1 third power plus 4

x3−4
x cubed minus 4

Answer 1

To find the inverse of the function \( f(x) = (x - 4)^{13} \), we need to follow these steps:

1. **Replace \( f(x) \) with \( y \)**:
\[
y = (x - 4)^{13}
\]

2. **Swap \( x \) and \( y \)**:
\[
x = (y - 4)^{13}
\]

3. **Solve for \( y \)**:
To solve for \( y \), we take the 13th root of both sides:
\[
y - 4 = x^{1/13}
\]
Then, solve for \( y \):
\[
y = x^{1/13} + 4
\]

4. **Write the inverse function**:
Thus, the inverse function \( f^{-1}(x) \) is:
\[
f^{-1}(x) = x^{1/13} + 4
\]

Now, looking at the options provided in your responses, we should evaluate which is correct:

- **f−1(x)=x³+4**: This is incorrect.
- **f−1(x)=(x−4)³**: This is incorrect.
- **x^{1/13}+4**: This is the correct form, but not listed exactly.
- **x³−4**: This is incorrect.

Thus, the correct inverse function is \( f^{-1}(x) = x^{1/13} + 4 \). However, it seems a direct match is not listed. Make sure to double-check the options if you need to select one based on formatting.