Question

If \( f(x) = \sqrt{x - 1 \) and \( g(f(x)) = x + 2\sqrt{x} + 1 \), then \( g(x) \) is:}

• A. \( (x+2)^2 \)
• B. \( (x-2)^2 \)
• C. \( (\sqrt{x}+2)^2 \)
• D. \( (\sqrt{x}-2)^2 \)

Hint:

To find \( g(x) \) when given \( g(f(x)) \), express \( x \) in terms of \( f(x) \) and substitute into \( g(f(x)) \).

Answer 1

The Correct Option is A

We are given the following functions: \[ f(x) = \sqrt{x} - 1 \] and \[ g(f(x)) = x + 2\sqrt{x} + 1 \] We need to find the function \( g(x) \). Step 1: Express \( f(x) \) and solve for \( x \). Since: \[ f(x) = \sqrt{x} - 1 \] we can rewrite it as: \[ \sqrt{x} = f(x) + 1 \] Squaring both sides: \[ x = (f(x) + 1)^2 \] Step 2: Substitute into the expression for \( g(f(x)) \). We are given: \[ g(f(x)) = x + 2\sqrt{x} + 1 \] Substitute \( x = (f(x) + 1)^2 \) and \( \sqrt{x} = f(x) + 1 \): \[ g(f(x)) = (f(x) + 1)^2 + 2(f(x) + 1) + 1 \] Step 3: Simplify the expression. Simplifying the expression, we get: \[ g(f(x)) = (f(x) + 1 + 2)^2 = (x + 2)^2 \] Thus, the final expression for \( g(x) \) is: \[ g(x) = (x + 2)^2 \] Final Answer: The correct option is ( \boxed{(x + 2)^2} \).