The problem provides two functions, $f(x) = 2x + 3$ and $g(x) = (x+2)^2 - 1$. We are asked to find (i) $f(g(x))$ and (ii) $g(f(1))$.

1. Problem Description

The problem provides two functions, $f(x) = 2x + 3$ and $g(x) = (x+2)^2 - 1$. We are asked to find (i) $f(g(x))$ and (ii) $g(f(1))$.

2. Solution Steps

(i) We need to find $f(g(x))$. This means we substitute $g(x)$ into $f(x)$.

$f(x) = 2x + 3$

$g(x) = (x+2)^2 - 1$

Therefore, $f(g(x)) = 2(g(x)) + 3 = 2((x+2)^2 - 1) + 3$.

Expanding the expression:

$f(g(x)) = 2(x^2 + 4x + 4 - 1) + 3 = 2(x^2 + 4x + 3) + 3 = 2x^2 + 8x + 6 + 3 = 2x^2 + 8x + 9$

(ii) We need to find $g(f(1))$. First, we find $f(1)$.

$f(1) = 2(1) + 3 = 2 + 3 = 5$

Now, we substitute $f(1)=5$ into $g(x)$ to find $g(f(1)) = g(5)$.

$g(5) = (5+2)^2 - 1 = (7)^2 - 1 = 49 - 1 = 48$

3. Final Answer

(i) $f(g(x)) = 2x^2 + 8x + 9$

(ii) $g(f(1)) = 48$