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.

Question

(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: \( f^{-1}(x) = \frac{-3x - 4}{x - 7} \).