Question

Which function is the inverse of f of x equals StartFraction x minus 5 over x minus 7 EndFraction, for x does not equal 7?

Answer options with 4 options
A.
f inverse of x equals StartFraction x minus 7 over x minus 5 EndFraction, for x does not equal 5

B.
f inverse of x equals StartFraction 5 x minus 1 over 7 x minus 1 EndFraction, for x does not equal StartFraction 1 over 7 EndFraction

C.
f inverse of x equals StartFraction 5 x minus 7 over x minus 1 EndFraction, for x does not equal 1

D.
f inverse of x equals StartFraction 7 x minus 5 over x minus 1 EndFraction, for x does not equal 1

Answer 1

To find the inverse of the function \( f(x) = \frac{x - 5}{x - 7} \), we need to follow these steps:

1. **Replace \( f(x) \) with \( y \)**:
\[
y = \frac{x - 5}{x - 7}
\]

2. **Swap \( x \) and \( y \)** to find the inverse:
\[
x = \frac{y - 5}{y - 7}
\]

3. **Solve for \( y \)**:
- Multiply both sides by \( (y - 7) \):
\[
x(y - 7) = y - 5
\]
- Distributing \( x \) gives:
\[
xy - 7x = y - 5
\]
- Rearranging terms to isolate \( y \):
\[
xy - y = 7x - 5
\]
- Factor \( y \) out:
\[
y(x - 1) = 7x - 5
\]
- Now, dividing both sides by \( (x - 1) \) gives:
\[
y = \frac{7x - 5}{x - 1}
\]

Thus, the inverse function is:
\[
f^{-1}(x) = \frac{7x - 5}{x - 1}
\]

Next, we also note the restriction for the inverse. Since \( y = \frac{7x - 5}{x - 1} \) will not be defined when \( x = 1 \).

Looking at the answer choices, we see:

**D.** \( f^{-1}(x) = \frac{7x - 5}{x - 1} \), for \( x \neq 1 \)

is the correct answer.

Thus, the answer is:
**D. \( f^{-1}(x) = \frac{7x - 5}{x - 1} \), for \( x \neq 1 \)**.