Question

Find the inverse of the function f left parenthesis x right parenthesis equals start fraction 7 x minus 4 over x plus 3 end fraction.
(1 point)
Responses

This function is not one-to-one and therefore has no inverse.
This function is not one-to-one and therefore has no inverse.

f superscript negative 1 baseline left parenthesis x right parenthesis equals start fraction negative 3 x minus 4 over x minus 7 end fraction
Image with alt text: f superscript negative 1 baseline left parenthesis x right parenthesis equals start fraction negative 3 x minus 4 over x minus 7 end fraction

f superscript negative 1 baseline left parenthesis x right parenthesis equals start fraction negative x plus 3 over 7 x minus 4 end fraction
Image with alt text: f superscript negative 1 baseline left parenthesis x right parenthesis equals start fraction negative x plus 3 over 7 x minus 4 end fraction

f superscript negative 1 baseline left parenthesis x right parenthesis equals start fraction 3 x plus 4 over x plus 7 end fraction

Answer 1

To find the inverse of the function \( f(x) = \frac{7x - 4}{x + 3} \), we will follow these steps:

1. Replace \( f(x) \) with \( y \):
\[
y = \frac{7x - 4}{x + 3}
\]

2. Swap \( x \) and \( y \) to find the inverse:
\[
x = \frac{7y - 4}{y + 3}
\]

3. Solve for \( y \). First, multiply both sides by \( y + 3 \) to eliminate the fraction:
\[
x(y + 3) = 7y - 4
\]
\[
xy + 3x = 7y - 4
\]

4. Rearrange the equation to isolate \( y \):
\[
xy - 7y = -4 - 3x
\]
\[
y(x - 7) = -4 - 3x
\]
\[
y = \frac{-4 - 3x}{x - 7}
\]

5. Thus, we have the inverse function:
\[
f^{-1}(x) = \frac{-3x - 4}{x - 7}
\]

Now, let's look at the options you provided:

- **This function is not one-to-one and therefore has no inverse.**
- **\( f^{-1}(x) = \frac{-3x - 4}{x - 7} \)**
- **\( f^{-1}(x) = \frac{-x + 3}{7x - 4} \)**
- **\( f^{-1}(x) = \frac{3x + 4}{x + 7} \)**

The correct answer is:
**\( f^{-1}(x) = \frac{-3x - 4}{x - 7} \)**.