To find the composite function $(g \circ f)(x)$, we apply the function $f(x)$ first and then apply the function $g(x)$ to the result of $f(x)$.

Given:

• $f(x) = x^2 - 4$
• $g(x) = x + 2$

We need to find $(g \circ f)(x)$, which is equivalent to $g(f(x))$.

Step-by-step solution:

1. First, find $f(x)$: $f(x) = x^2 - 4$

2. Now, apply $g(x)$ to $f(x)$, which means we substitute $f(x)$ into $g(x)$: $g(f(x)) = g(x^2 - 4)$ Since $g(x) = x + 2$, we replace $x$ with $x^2 - 4$: $g(x^2 - 4) = (x^2 - 4) + 2$

3. Simplify: $g(f(x)) = x^2 - 4 + 2 = x^2 - 2$

So, the composite function $(g \circ f)(x)$ is: $(g \circ f)(x) = x^2 - 2$

Would you like more details or have any further questions?

Here are 5 related questions to explore:

1. How do you find the composite function $(f \circ g)(x)$?
2. What is the domain of the composite function $(g \circ f)(x)$?
3. How would the composite function change if $g(x) = 2x + 3$?
4. Can you graph the composite function $(g \circ f)(x) = x^2 - 2$?
5. How would you reverse the process and find the individual functions from a given composite function?

Tip: When working with composite functions, always apply the inner function first before applying the outer function!