Question

If \(g(x)\) is the inverse function of \(f(x)\), where \(f(x) = \frac{5x+3}{4x-1}\), then \(g(1) =\)

• A. \(-4\)
• B. \(\frac{-1}{4}\)
• C. \(\frac{8}{3}\)
• D. \(\frac{3}{8}\)

Hint:

\(g(1)\) is the number \(x\) for which \(f(x) = 1\).

Answer 1

The Correct Option is A

Step 1: Plan:
Find the full inverse and then put $x = 1$.

Step 2: Inverse:
Let $y = \frac{5x+3}{4x-1}$. Then $4xy - y = 5x + 3$, so $x(4y-5) = y + 3$ and $x = \frac{y+3}{4y-5}$. So $g(y) = \frac{y+3}{4y-5}$.
$g(1) = \frac{4}{-1} = -4$.

Final Answer:
$g(1) = -4$, option (A). \[ \boxed{-4} \]