QuestionFor $f(x) = x^2$ and $g(x) = x^2 + 4$ , find the following composite functions and state the domain of each.
(a) $f \circ g$ (b) $g \circ f$ (c) $f \circ f$ (d) $g \circ g$

Studdy Solution

STEP 1

What is this asking? We need to find four composite functions, which basically means plugging one function into another, like a Russian nesting doll, and then figure out what values of $x$ are allowed. Watch out! Don't mix up the order of the functions when composing them! $f \circ g$ means $f(g(x))$ , not $g(f(x))$ .
Also, be careful when finding the domain; think about what values of $x$ would cause any problems, like dividing by zero or taking the square root of a negative number.

STEP 2

1. Find $f \circ g$ and its domain.
2. Find $g \circ f$ and its domain.
3. Find $f \circ f$ and its domain.
4. Find $g \circ g$ and its domain.

STEP 3

Alright, let's start with $f \circ g$ , which means $f(g(x))$ .
This is like putting the $g(x)$ function inside the $f(x)$ function!

STEP 4

We know $f(x) = x^2$ and $g(x) = x^2 + 4$ .
So, $f(g(x))$ means we replace the $x$ in $f(x)$ with the entire $g(x)$ function.

STEP 5

So, $f(g(x)) = (g(x))^2 = (x^2 + 4)^2$ .
Let's expand this: $(x^2 + 4)^2 = x^4 + 8x^2 + 16$ .

STEP 6

Now, for the domain.
Since we're only dealing with polynomials, there are no restrictions on $x$ . $x$ can be anything!
So, the domain of $f \circ g$ is all real numbers, which we can write as $(-\infty, \infty)$ .

STEP 7

Next up: $g \circ f$ , which means $g(f(x))$ .
This time, we're putting $f(x)$ inside $g(x)$ .

STEP 8

We have $g(x) = x^2 + 4$ , so $g(f(x)) = (f(x))^2 + 4$ .
Since $f(x) = x^2$ , we get $g(f(x)) = (x^2)^2 + 4 = x^4 + 4$ .

STEP 9

Again, we have a polynomial, so the domain is all real numbers: $(-\infty, \infty)$ .
No restrictions here!

STEP 10

Now for $f \circ f$ , meaning $f(f(x))$ .
We're putting $f(x)$ inside itself!
It's like function inception!

STEP 11

We have $f(x) = x^2$ , so $f(f(x)) = (f(x))^2 = (x^2)^2 = x^4$ .

STEP 12

And, as you might expect, the domain is still all real numbers: $(-\infty, \infty)$ .

STEP 13

Finally, let's tackle $g \circ g$ , which is $g(g(x))$ .
Putting $g(x)$ inside itself!

STEP 14

We have $g(x) = x^2 + 4$ , so $g(g(x)) = (g(x))^2 + 4 = (x^2 + 4)^2 + 4$ .

STEP 15

Expanding this, we get $(x^2 + 4)^2 + 4 = x^4 + 8x^2 + 16 + 4 = x^4 + 8x^2 + 20$ .

STEP 16

Once again, a polynomial!
So, the domain is all real numbers: $(-\infty, \infty)$ .

STEP 17

(a) $f \circ g = x^4 + 8x^2 + 16$ , Domain: $(-\infty, \infty)$ (b) $g \circ f = x^4 + 4$ , Domain: $(-\infty, \infty)$ (c) $f \circ f = x^4$ , Domain: $(-\infty, \infty)$ (d) $g \circ g = x^4 + 8x^2 + 20$ , Domain: $(-\infty, \infty)$

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