To solve this problem, we need to find the composition of two functions $f$ and $g$, denoted as $(f \circ g)(x)$, which represents applying $g(x)$ first and then applying $f$ to the result of $g(x)$.

Given Functions

• $f(x) = 5x - 2$
• $g(x) = x^2$

Composition $(f \circ g)(x)$

To find $(f \circ g)(x)$, we substitute $g(x)$ into $f(x)$: $(f \circ g)(x) = f(g(x))$ Since $g(x) = x^2$, this becomes: $f(g(x)) = f(x^2)$ Now, substitute $x^2$ into the function $f(x)$: $f(x^2) = 5(x^2) - 2$ Simplify this expression: $f(x^2) = 5x^2 - 2$

Answer

The correct answer is:

• Option D: $5x^2 - 2$

Would you like a further breakdown of any step or a detailed explanation?

Here are some related questions to practice function composition and function properties:

1. What is the result of $(g \circ f)(x)$ using the same functions $f(x) = 5x - 2$ and $g(x) = x^2$?
2. How would you find $(f \circ f)(x)$ for $f(x) = 5x - 2$?
3. Can function composition be commutative, i.e., is $(f \circ g)(x) = (g \circ f)(x)$ always true?
4. If $f(x) = ax + b$ and $g(x) = cx + d$, how do you compute $(f \circ g)(x)$ in terms of $a$, $b$, $c$, and $d$?
5. If $f(x)$ and $g(x)$ are both quadratic functions, is the composition $(f \circ g)(x)$ necessarily quadratic?

Tip: When composing functions, always apply the inner function first and then substitute the result into the outer function carefully to avoid errors.