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: {Find \( f^{-1}(3) \)}
\[ y = f(x) = 5x - 3. \] \[ x = \frac{y + 3}{5}. \] \[ f^{-1}(3) = \frac{6}{5}. \] Step 2: {Compute \( g(f^{-1}(3)) \)}
\[ g(x) = x^2 + 3. \] \[ g \left( \frac{6}{5} \right) = \left( \frac{6}{5} \right)^2 + 3. \] \[ = \frac{36}{25} + 3 = \frac{111}{25}. \] Step 3: {Conclusion}
Thus, \( g \circ f^{-1}(3) = \frac{111}{25} \).

Answer 2

Step 1: Find the inverse function \( f^{-1}(x) \)
We are given \( f(x) = 5x - 3 \). To find the inverse, we solve for \( x \) in terms of \( y \): \[ y = 5x - 3 \quad \Rightarrow \quad y + 3 = 5x \quad \Rightarrow \quad x = \frac{y + 3}{5}. \] Thus, the inverse function is: \[ f^{-1}(x) = \frac{x + 3}{5}. \] Step 2: Calculate \( f^{-1}(3) \)
Substitute \( x = 3 \) into the inverse function: \[ f^{-1}(3) = \frac{3 + 3}{5} = \frac{6}{5}. \] Step 3: Evaluate \( g \circ f^{-1}(3) \)
We are given \( g(x) = x^2 + 3 \), so we now evaluate \( g \) at \( f^{-1}(3) \), which is \( \frac{6}{5} \): \[ g\left( \frac{6}{5} \right) = \left( \frac{6}{5} \right)^2 + 3 = \frac{36}{25} + 3 = \frac{36}{25} + \frac{75}{25} = \frac{111}{25}. \] Final Answer:
The value of \( g \circ f^{-1}(3) \) is:

\( \frac{111}{25} \)