Question

Let \(f(x)=\frac{4x+3}{x+2}\), Then the value of \(f^{-1}(-2)\) is equal to

• A. \(\frac{7}{5}\)
• B. \(\frac{-7}{6}\)
• C. \(\frac{-7}{5}\)
• D. \(\frac{7}{6}\)
• E. \(\frac{5}{6}\)

Answer 1

The Correct Option is B

We are given the function:

\[ f(x) = \frac{4x + 3}{x + 2} \]

To find \( f^{-1}(-2) \), we need to find the inverse of the function and then substitute \( -2 \) for \( y \).

Let \( y = f(x) = \frac{4x + 3}{x + 2} \).

Now solve for \( x \) in terms of \( y \):

\[ y(x + 2) = 4x + 3 \] \[ yx + 2y = 4x + 3 \] \[ yx - 4x = 3 - 2y \] \[ x(y - 4) = 3 - 2y \] \[ x = \frac{3 - 2y}{y - 4} \]

This is the inverse function:

\[ f^{-1}(y) = \frac{3 - 2y}{y - 4} \]

Now substitute \( y = -2 \) into the inverse function:

\[ f^{-1}(-2) = \frac{3 - 2(-2)}{-2 - 4} = \frac{3 + 4}{-6} = \frac{7}{-6} = \frac{-7}{6} \]

Answer: \( \frac{-7}{6} \)