Question

If \( f: \mathbb{R} \to \mathbb{R} \), \( g: \mathbb{R} \to \mathbb{R} \) are defined by \( f(x) = 5x - 3 \), \( g(x) = x^2 + 3 \), then \( g \circ f^{-1}(3) \) is equal to

• A. \( \frac{25}{3} \)
• B. \( \frac{111}{25} \)
• C. \( \frac{9}{25} \)
• D. \( \frac{25}{111} \)

Hint:

To find \( g \circ f^{-1} \), first determine \( f^{-1}(x) \), then substitute into \( g(x) \).

Answer 1

The Correct Option is B

Step 1: Determine the value of \( f^{-1}(3) \).
\[y = f(x) = 5x - 3.\]\[x = \frac{y + 3}{5}.\]\[f^{-1}(3) = \frac{6}{5}.\]Step 2: Calculate \( g(f^{-1}(3)) \).
\[g(x) = x^2 + 3.\]\[g \left( \frac{6}{5} \right) = \left( \frac{6}{5} \right)^2 + 3.\]\[= \frac{36}{25} + 3 = \frac{111}{25}.\]Step 3: State the final result.
Therefore, \( g \circ f^{-1}(3) = \frac{111}{25} \).