We are tasked with finding $(f \circ g)(x)$, which represents the composition of the functions $f(x)$ and $g(x)$.

Given:

• $f(x) = 2x + 3$
• $g(x) = x^2$

Steps to find $(f \circ g)(x)$:

1. By definition, $(f \circ g)(x) = f(g(x))$.
2. Substitute $g(x)$ into $f(x)$. That means replacing the $x$ in $f(x)$ with $g(x)$.

So: $(f \circ g)(x) = f(g(x)) = f(x^2)$

3. Now, substitute $x^2$ into $f(x) = 2x + 3$:

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

4. Simplify the expression:

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

Thus, $(f \circ g)(x) = 2x^2 + 3$.

Would you like further clarification or more details on function compositions?

Related questions:

1. How do you find the composition of three functions?
2. What are the domain and range of $f \circ g$ given the functions $f(x) = 2x + 3$ and $g(x) = x^2$?
3. How does the order of composition affect the result, i.e., $f(g(x))$ versus $g(f(x))$?
4. What is the graphical interpretation of the composition of two functions?
5. Can a composition of functions ever be undefined, and under what conditions?

Tip: Always check the domain of each function before composing to ensure the composition is valid for all $x$.