QuestionFor each function $h$ , find functions $f$ and $g$ so that $h(x)=(f \circ g)(x)$ .
10. $h(x)=\left|x^{5}-3\right|$

Studdy Solution

STEP 1

What is this asking? We need to find two functions, $f$ and $g$ , that when we plug $g(x)$ into $f(x)$ , it gives us $h(x)$ , which is the absolute value of $x^5 - 3$ . Watch out! There might be more than one correct answer, so don't be afraid to get creative!
Just make sure $f$ and $g$ work together to give us $h$ .

STEP 2

1. Break It Down
2. Define the functions

STEP 3

Alright, let's look at our $h(x)$ .
We've got $h(x) = |x^5 - 3|$ .
Notice how we're taking the absolute value of something, right?
That "something" is $x^5 - 3$ .

STEP 4

So, we're doing two things here: first, we're taking $x$ to the fifth power and subtracting 3, and then we're taking the absolute value of the result.
That's our clue for how to define $f$ and $g$ !

STEP 5

Let's define $g(x)$ to be the inside part: $g(x) = x^5 - 3$ So, $g(x)$ takes $x$ , raises it to the fifth power, and then subtracts 3.
Awesome!

STEP 6

Now, let's define $f(x)$ to be the absolute value function: $f(x) = |x|$ So $f(x)$ takes whatever we give it and makes it positive!
Perfect!

STEP 7

Now, let's check if our $f(x)$ and $g(x)$ work together.
We want to find $(f \circ g)(x)$ , which means we plug $g(x)$ into $f(x)$ .

STEP 8

Let's do it!
We have $f(x) = |x|$ , and we're plugging in $g(x)$ for $x$ .
So, we get: $(f \circ g)(x) = f(g(x)) = f(x^5 - 3) = |x^5 - 3|$ Boom! That's exactly our original $h(x)$ !
We did it!

STEP 9

We found our functions! $f(x) = |x|$ and $g(x) = x^5 - 3$ .
When we combine them, we get $(f \circ g)(x) = h(x) = |x^5 - 3|$ .

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