To find $(f \circ g)(x)$, also written as $f(g(x))$, we substitute the expression for $g(x)$ into $f(x)$.

Step 1: Write out $f(g(x))$

We are given:

• $f(x) = 2x + 5$
• $g(x) = 4x^2 + 3x$

Now substitute $g(x)$ into $f(x)$: $f(g(x)) = f(4x^2 + 3x)$

Step 2: Apply $f(x) = 2x + 5$ to $g(x)$

To do this, replace $x$ in $f(x)$ with $4x^2 + 3x$: $f(g(x)) = 2(4x^2 + 3x) + 5$

Step 3: Simplify the expression

Distribute the 2 across the terms inside the parentheses: $f(g(x)) = 2(4x^2) + 2(3x) + 5$ $f(g(x)) = 8x^2 + 6x + 5$

So, the simplified form of $(f \circ g)(x)$ is: $(f \circ g)(x) = 8x^2 + 6x + 5$

Would you like more details or have any questions?

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Here are 5 related questions:

1. How would you compute $(g \circ f)(x)$?
2. What is the domain of $(f \circ g)(x)$?
3. Can you graph the function $(f \circ g)(x)$?
4. How do the graphs of $f(x)$ and $g(x)$ differ?
5. What happens if $f(x)$ is a constant function instead?

Tip: When composing functions, always carefully substitute one function into the other and simplify step by step.