Question

1. if $f(x) = 2x + 4$ and $g(f(x)) = \sqrt{2x + 5}$, what is the equation for $g(x)$?

a) $g(x) = \sqrt{x}$
b) $g(x) = \sqrt{x + 1}$
c) $g(x) = \sqrt{2x + 5}$
d) $g(x) = \sqrt{2x + 1}$

Explanation:

Step1: Let \( y = f(x) \)

Given \( f(x)=2x + 4 \), so \( y=2x + 4 \). We can solve for \( x \) in terms of \( y \): \( 2x=y - 4\), then \( x=\frac{y - 4}{2} \).

Step2: Substitute \( x=\frac{y - 4}{2} \) into \( g(f(x))=\sqrt{2x + 5} \)

Since \( g(f(x)) = g(y) \) (because \( y = f(x) \)), substitute \( x=\frac{y - 4}{2} \) into \( \sqrt{2x + 5} \):
\[

$$\begin{align*} g(y)&=\sqrt{2\times\frac{y - 4}{2}+5}\\ &=\sqrt{(y - 4)+5}\\ &=\sqrt{y+1} \end{align*}$$

\]
Since the variable is a dummy variable, we can replace \( y \) with \( x \), so \( g(x)=\sqrt{x + 1} \).

Answer:

B) \( g(x)=\sqrt{x + 1} \)