Math Snap

STEP 1

What is this asking?
We need to find the composition of two functions, $f(x)$ and $g(x)$, both when $x$ is a variable and when $x$ is the number $4$.
Watch out!
Function composition isn't just multiplying the functions!
It's about plugging one function into the other.

STEP 2

1. Find $(f \circ g)(x)$
2. Find $(f \circ g)(4)$

STEP 3

Alright, let's start with what $(f \circ g)(x)$ even means.
It means we take $g(x)$ and plug it in wherever we see an $x$ in $f(x)$.
It's like a function turducken!

STEP 4

We know that $f(x) = x^2 + 3$ and $g(x) = 5x - 3$.
So, $(f \circ g)(x)$ becomes $f(g(x))$.
Let's substitute $g(x)$ into $f(x)$:
$$f(g(x)) = (g(x))^2 + 3$$

STEP 5

Now, let's replace $g(x)$ with its actual definition, which is $5x - 3$:
$$(5x - 3)^2 + 3$$

STEP 6

Time to expand that square!
Remember $(a - b)^2 = a^2 - 2ab + b^2$.
Here, $a = 5x$ and $b = 3$, so we get:
$$(5x)^2 - 2 \cdot (5x) \cdot 3 + 3^2 + 3$$

STEP 7

Simplify that expression:
$$25x^2 - 30x + 9 + 3$$ $$25x^2 - 30x + 12$$

STEP 8

Now that we have a nice and simplified expression for $(f \circ g)(x)$, which is $25x^2 - 30x + 12$, we can easily find $(f \circ g)(4)$ by substituting $x = 4$.

STEP 9

Let's plug in $x = 4$:
$$25 \cdot (4)^2 - 30 \cdot 4 + 12$$

STEP 10

Calculate the square:
$$25 \cdot 16 - 30 \cdot 4 + 12$$

STEP 11

Perform the multiplications:
$$400 - 120 + 12$$

STEP 12

Combine the terms:
$$280 + 12$$ $$292$$

a. $(f \circ g)(x) = 25x^2 - 30x + 12$
b. $(f \circ g)(4) = 292$