To figure out composite functions step by step, it’s important to know what they are. A composite function connects two functions, like $f(x)$ and $g(x)$.

In simple terms, the composite function $f(g(x))$ starts with an input $x$. First, you apply $g$ to that input, then you take what $g$ gives you and use it as the input for $f$.

Let’s break it down into easy-to-follow steps:

1. Identify the Functions: First, figure out what the two functions are. For example:

   • $f(x) = 2x + 3$
   • $g(x) = x^2$

2. Decide What to Find: It’s important to know which composite function you want to calculate. Here, we can work with $f(g(x))$ and $g(f(x))$. Let’s start with $f(g(x))$.

3. Substitute: For $f(g(x))$, you replace $x$ in $f(x)$ with $g(x)$. This means we are finding $f(g(x))$ by putting $g(x)$ into $f(x)$:

   • Since $g(x) = x^2$, replace this in $f(x)$: $f(g(x)) = f(x^2)$

4. Apply the Outer Function: Now we perform the operation defined by $f$. We know that: $f(x) = 2x + 3$ So: $f(x^2) = 2(x^2) + 3 = 2x^2 + 3$

5. Result of the Composite: Now we have: $f(g(x)) = 2x^2 + 3$

Next, let’s work on $g(f(x))$.

1. Substitute Again: Now we compute $g(f(x))$. This time, replace $f(x)$ into $g(x)$:

   • Since $f(x) = 2x + 3$, we want to find $g(f(x)) = g(2x + 3)$.

2. Apply the Function: Now evaluate: $g(x) = x^2$ Therefore: $g(2x + 3) = (2x + 3)^2$

3. Expand: To get the final answer, expand: $(2x + 3)^2 = 4x^2 + 12x + 9$

4. Result of the Second Composite: So we find: $g(f(x)) = 4x^2 + 12x + 9$

In summary, figuring out composite functions includes:

• Identifying the functions you are working with.
• Deciding which composite function to calculate.
• Substituting the inner function into the outer function.
• Doing the math needed for the outer function.
• Simplifying to find your final answer.

Composite functions help you see how different functions connect, which can be really useful in real-life situations. The more you practice with these steps and different functions, the easier they will become!