Solved on Jan 23, 2024

Evaluate $(f \circ g)(1), (f \circ g)(-1), (g \circ f)(-1), (g \circ f)(0), (g \circ g)(-2), (f \circ f)(-1)$ using the given values of $\mathbf{f}(\mathbf{x})$ and $\mathbf{g}(\mathbf{x})$ .

STEP 1

Assumptions
1. The function $f(x)$ and $g(x)$ are defined by the values given in the table.
2. The notation $(f \circ g)(x)$ means the composition of $f$ and $g$ , which is $f(g(x))$ .
3. The notation $(g \circ f)(x)$ means the composition of $g$ and $f$ , which is $g(f(x))$ .
4. We will use the table to find the values of $g(x)$ and $f(x)$ and then use these values to find the compositions.

STEP 2

To find $(f \circ g)(1)$ , we first find $g(1)$ and then find $f(g(1))$ .

STEP 3

Look up the value of $g(1)$ in the table.
$g(1) = 0$

STEP 4

Now find $f(g(1))$ which is $f(0)$ .

STEP 5

Look up the value of $f(0)$ in the table.
$f(0) = -1$

STEP 6

Therefore, $(f \circ g)(1) = -1$ .

STEP 7

To find $(f \circ g)(-1)$ , we first find $g(-1)$ and then find $f(g(-1))$ .

STEP 8

Look up the value of $g(-1)$ in the table.
$g(-1) = 0$

STEP 9

Now find $f(g(-1))$ which is $f(0)$ .

STEP 10

We have already found $f(0)$ in STEP_5.
$f(0) = -1$

STEP 11

Therefore, $(f \circ g)(-1) = -1$ .

STEP 12

To find $(g \circ f)(-1)$ , we first find $f(-1)$ and then find $g(f(-1))$ .

STEP 13

Look up the value of $f(-1)$ in the table.
$f(-1) = -3$

STEP 14

Now find $g(f(-1))$ which is $g(-3)$ .

STEP 15

Look up the value of $g(-3)$ in the table.
$g(-3) = 7$

STEP 16

Therefore, $(g \circ f)(-1) = 7$ .

STEP 17

To find $(g \circ f)(0)$ , we first find $f(0)$ and then find $g(f(0))$ .

STEP 18

We have already found $f(0)$ in STEP_5.
$f(0) = -1$

STEP 19

Now find $g(f(0))$ which is $g(-1)$ .

STEP 20

Look up the value of $g(-1)$ in the table.
$g(-1) = 0$

STEP 21

Therefore, $(g \circ f)(0) = 0$ .

STEP 22

To find $(g \circ g)(-2)$ , we first find $g(-2)$ and then find $g(g(-2))$ .

STEP 23

Look up the value of $g(-2)$ in the table.
$g(-2) = 2$

STEP 24

Now find $g(g(-2))$ which is $g(2)$ .

STEP 25

Look up the value of $g(2)$ in the table.
$g(2) = 2$

STEP 26

Therefore, $(g \circ g)(-2) = 2$ .

STEP 27

To find $(f \circ f)(-1)$ , we first find $f(-1)$ and then find $f(f(-1))$ .

STEP 28

We have already found $f(-1)$ in STEP_13.
$f(-1) = -3$

STEP 29

Now find $f(f(-1))$ which is $f(-3)$ .

STEP 30

Look up the value of $f(-3)$ in the table.
$f(-3) = -5$
Therefore, the evaluated expressions are: a. $(f \circ g)(1) = -1$ b. $(f \circ g)(-1) = -1$ c. $(g \circ f)(-1) = 7$ d. $(g \circ f)(0) = 0$ e. $(g \circ g)(-2) = 2$ f. $(f \circ f)(-1) = -5$

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STEP 1

STEP 2

To find (f∘g)(1)(f \circ g)(1)(f∘g)(1), we first find g(1)g(1)g(1) and then find f(g(1))f(g(1))f(g(1)).

STEP 3

Look up the value of g(1)g(1)g(1) in the table.g(1)=0g(1) = 0g(1)=0

STEP 4

Now find f(g(1))f(g(1))f(g(1)) which is f(0)f(0)f(0).

STEP 5

Look up the value of f(0)f(0)f(0) in the table.f(0)=−1f(0) = -1f(0)=−1

STEP 6

Therefore, (f∘g)(1)=−1(f \circ g)(1) = -1(f∘g)(1)=−1.

STEP 7

To find (f∘g)(−1)(f \circ g)(-1)(f∘g)(−1), we first find g(−1)g(-1)g(−1) and then find f(g(−1))f(g(-1))f(g(−1)).

STEP 8

Look up the value of g(−1)g(-1)g(−1) in the table.g(−1)=0g(-1) = 0g(−1)=0

STEP 9

Now find f(g(−1))f(g(-1))f(g(−1)) which is f(0)f(0)f(0).

STEP 10

We have already found f(0)f(0)f(0) in STEP_5.f(0)=−1f(0) = -1f(0)=−1

STEP 11

Therefore, (f∘g)(−1)=−1(f \circ g)(-1) = -1(f∘g)(−1)=−1.

STEP 12

To find (g∘f)(−1)(g \circ f)(-1)(g∘f)(−1), we first find f(−1)f(-1)f(−1) and then find g(f(−1))g(f(-1))g(f(−1)).

STEP 13

Look up the value of f(−1)f(-1)f(−1) in the table.f(−1)=−3f(-1) = -3f(−1)=−3

STEP 14

Now find g(f(−1))g(f(-1))g(f(−1)) which is g(−3)g(-3)g(−3).

STEP 15

Look up the value of g(−3)g(-3)g(−3) in the table.g(−3)=7g(-3) = 7g(−3)=7

STEP 16

Therefore, (g∘f)(−1)=7(g \circ f)(-1) = 7(g∘f)(−1)=7.

STEP 17

To find (g∘f)(0)(g \circ f)(0)(g∘f)(0), we first find f(0)f(0)f(0) and then find g(f(0))g(f(0))g(f(0)).

STEP 18

We have already found f(0)f(0)f(0) in STEP_5.f(0)=−1f(0) = -1f(0)=−1

STEP 19

Now find g(f(0))g(f(0))g(f(0)) which is g(−1)g(-1)g(−1).

STEP 20

Look up the value of g(−1)g(-1)g(−1) in the table.g(−1)=0g(-1) = 0g(−1)=0

STEP 21

Therefore, (g∘f)(0)=0(g \circ f)(0) = 0(g∘f)(0)=0.

STEP 22

To find (g∘g)(−2)(g \circ g)(-2)(g∘g)(−2), we first find g(−2)g(-2)g(−2) and then find g(g(−2))g(g(-2))g(g(−2)).

STEP 23

Look up the value of g(−2)g(-2)g(−2) in the table.g(−2)=2g(-2) = 2g(−2)=2

STEP 24

Now find g(g(−2))g(g(-2))g(g(−2)) which is g(2)g(2)g(2).

STEP 25

Look up the value of g(2)g(2)g(2) in the table.g(2)=2g(2) = 2g(2)=2

STEP 26

Therefore, (g∘g)(−2)=2(g \circ g)(-2) = 2(g∘g)(−2)=2.

STEP 27

To find (f∘f)(−1)(f \circ f)(-1)(f∘f)(−1), we first find f(−1)f(-1)f(−1) and then find f(f(−1))f(f(-1))f(f(−1)).

STEP 28

We have already found f(−1)f(-1)f(−1) in STEP_13.f(−1)=−3f(-1) = -3f(−1)=−3

STEP 29

Now find f(f(−1))f(f(-1))f(f(−1)) which is f(−3)f(-3)f(−3).

STEP 30

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To find $(f \circ g)(1)$ , we first find $g(1)$ and then find $f(g(1))$ .

Look up the value of $g(1)$ in the table.
$g(1) = 0$

Now find $f(g(1))$ which is $f(0)$ .

Look up the value of $f(0)$ in the table.
$f(0) = -1$

Therefore, $(f \circ g)(1) = -1$ .

To find $(f \circ g)(-1)$ , we first find $g(-1)$ and then find $f(g(-1))$ .

Look up the value of $g(-1)$ in the table.
$g(-1) = 0$

Now find $f(g(-1))$ which is $f(0)$ .

We have already found $f(0)$ in STEP_5.
$f(0) = -1$

Therefore, $(f \circ g)(-1) = -1$ .

To find $(g \circ f)(-1)$ , we first find $f(-1)$ and then find $g(f(-1))$ .

Look up the value of $f(-1)$ in the table.
$f(-1) = -3$

Now find $g(f(-1))$ which is $g(-3)$ .

Look up the value of $g(-3)$ in the table.
$g(-3) = 7$

Therefore, $(g \circ f)(-1) = 7$ .

To find $(g \circ f)(0)$ , we first find $f(0)$ and then find $g(f(0))$ .

We have already found $f(0)$ in STEP_5.
$f(0) = -1$

Now find $g(f(0))$ which is $g(-1)$ .

Look up the value of $g(-1)$ in the table.
$g(-1) = 0$

Therefore, $(g \circ f)(0) = 0$ .

To find $(g \circ g)(-2)$ , we first find $g(-2)$ and then find $g(g(-2))$ .

Look up the value of $g(-2)$ in the table.
$g(-2) = 2$

Now find $g(g(-2))$ which is $g(2)$ .

Look up the value of $g(2)$ in the table.
$g(2) = 2$

Therefore, $(g \circ g)(-2) = 2$ .

To find $(f \circ f)(-1)$ , we first find $f(-1)$ and then find $f(f(-1))$ .

We have already found $f(-1)$ in STEP_13.
$f(-1) = -3$

Now find $f(f(-1))$ which is $f(-3)$ .