Math / Algebra

Question6. Consider the tables of values for the two functions shown. \begin{tabular}{|c|c|c|} \hline $x$ & $y=f(x)$ & $y=g(x)$ \\ \hline-2 & 5 & -2 \\ \hline-1 & 6 & 1 \\ \hline 0 & 8 & 2 \\ \hline 1 & 7 & 0 \\ \hline 2 & 9 & -1 \\ \hline \end{tabular}
Complete the table of values for the composite function $y=f(g(x))$ . \begin{tabular}{|c|c|} \hline $x$ & $y=f(g(x))$ \\ \hline-2 & 5 \\ \hline-1 & 7 \\ \hline 0 & 9 \\ \hline 1 & 8 \\ \hline 2 & 6 \\ \hline \end{tabular}

Studdy Solution

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!

STEP 13

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

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Studdy Solution

STEP 1

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!

STEP 13

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

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Studdy Solution

STEP 1

STEP 2

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

STEP 3

Let's **start** with x=−2x = -2x=−2.From the table, we see that g(−2)=−2g(-2) = -2g(−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=−1x = -1x=−1.We see g(−1)=1g(-1) = 1g(−1)=1.Awesome!

STEP 5

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

STEP 6

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

STEP 7

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

STEP 8

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

STEP 9

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

STEP 10

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

STEP 11

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

STEP 12

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

STEP 13

Here's the completed table for y=f(g(x))y = f(g(x))y=f(g(x)):xy=f(g(x))−25−17091826\begin{array}{cc} x & y=f(g(x)) \\ -2 & 5 \\ -1 & 7 \\ 0 & 9 \\ 1 & 8 \\ 2 & 6 \\ \end{array}x−2−1012​y=f(g(x))57986​

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

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!

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

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

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

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

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!

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

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

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

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 $y = f(g(x))$ :
$\begin{array}{cc} x & y=f(g(x)) \\ -2 & 5 \\ -1 & 7 \\ 0 & 9 \\ 1 & 8 \\ 2 & 6 \\ \end{array}$