Math Snap

STEP 1

What is this asking?
We're given two functions, $f(x)$ and $g(x)$, as tables of values, and we need to find the values of the composite function $f(g(x))$ for some given $x$ values.
Watch out!
Remember that $f(g(x))$ means we first apply $g(x)$, and then apply $f(x)$ to the result.
It's like a two-step process!
Don't mix up the order.

STEP 2

1. Evaluate $g(x)$
2. Evaluate $f(g(x))$

STEP 3

Let's start with $x = -2$.
From the table, we see that $g(-2) = -2$.
This is our inner function result.
Keep this value in mind as we move to the next sub-step!

STEP 4

Now, let's do $x = -1$.
We see $g(-1) = 1$.
Awesome!

STEP 5

For $x = 0$, we have $g(0) = 2$.
See how we're just looking up the values in the $g(x)$ table?

STEP 6

Next, for $x = 1$, we find $g(1) = 0$.
Almost there!

STEP 7

Finally, for $x = 2$, we have $g(2) = -1$.
We've found all the inner function values!

STEP 8

Remember that $g(-2) = -2$?
So, $f(g(-2))$ is the same as $f(-2)$.
Looking at the table for $f(x)$, we find $f(-2) = 5$.
So, $f(g(-2)) = 5$.
Boom!

STEP 9

We found $g(-1) = 1$, so we need $f(1)$.
The table tells us $f(1) = 7$, so $f(g(-1)) = 7$.
Fantastic!

STEP 10

We have $g(0) = 2$, so we need $f(2)$.
The table says $f(2) = 9$.
Thus, $f(g(0)) = 9$.
Keep it going!

STEP 11

Since $g(1) = 0$, we want $f(0)$.
We see $f(0) = 8$, which means $f(g(1)) = 8$.
Almost done!

STEP 12

Lastly, we know $g(2) = -1$, so we need $f(-1)$.
The table gives us $f(-1) = 6$, so $f(g(2)) = 6$.
We did it!

Here's the completed table for $y = f(g(x))$:
$$\begin{array}{cc} x & y=f(g(x)) \\ -2 & 5 \\ -1 & 7 \\ 0 & 9 \\ 1 & 8 \\ 2 & 6 \\ \end{array}$$