Question

Let \[ g(x) = 4x + 3 \quad {and} \quad f(g(x)) = x^2 + 9. \] Then the value of \( f(7) \) is equal to

• A. \( 7 \)
• B. \( 9 \)
• C. \( 10 \)
• D. \( 12 \)
• E. \( 14 \)

Hint:

For nested functions, express \( x \) in terms of \( g(x) \) and substitute it in \( f(x) \).

Answer 1

The Correct Option is C

We are given: \[ f(g(x)) = x^2 + 9 \] Substituting \( g(x) = 4x + 3 \), we get: \[ f(4x + 3) = x^2 + 9 \] Replacing \( x \) in terms of \( g(x) \), \[ x = \frac{g(x) - 3}{4} \] Thus, \[ f(y) = \left( \frac{y - 3}{4} \right)^2 + 9 \] Now, substituting \( y = 7 \): \[ f(7) = \left( \frac{7 - 3}{4} \right)^2 + 9 \] \[ = \left( \frac{4}{4} \right)^2 + 9 \] \[ = 1 + 9 = 10 \] Thus, the correct answer is (C).