Math / Algebra

QuestionFind $(f \circ g)(x)$ and $(g \circ f)(x)$ and the domain of each. $f(x)=x+17, g(x)=x-17$ $(f \circ g)(x)=$ $\square$ (Simplify your answer.)

Studdy Solution

STEP 1

1. We are given two functions: $f(x) = x + 17$ and $g(x) = x - 17$ .
2. We need to find the compositions $(f \circ g)(x)$ and $(g \circ f)(x)$ .
3. The domain of a composition of functions is the set of all $x$ such that the composition is defined.

STEP 2

1. Find the composition $(f \circ g)(x)$ .
2. Simplify $(f \circ g)(x)$ .
3. Find the domain of $(f \circ g)(x)$ .
4. Find the composition $(g \circ f)(x)$ .
5. Simplify $(g \circ f)(x)$ .
6. Find the domain of $(g \circ f)(x)$ .

STEP 3

To find $(f \circ g)(x)$ , we substitute $g(x)$ into $f(x)$ . This means we replace every $x$ in $f(x)$ with $g(x)$ .
$(f \circ g)(x) = f(g(x)) = f(x - 17)$

STEP 4

Now, simplify $f(x - 17)$ using the definition of $f(x)$ .
$f(x - 17) = (x - 17) + 17$
Simplify the expression:
$(x - 17) + 17 = x$
Thus, $(f \circ g)(x) = x$ .

STEP 5

The domain of $(f \circ g)(x) = x$ is the set of all real numbers because there are no restrictions on $x$ .

STEP 6

To find $(g \circ f)(x)$ , we substitute $f(x)$ into $g(x)$ . This means we replace every $x$ in $g(x)$ with $f(x)$ .
$(g \circ f)(x) = g(f(x)) = g(x + 17)$

STEP 7

Now, simplify $g(x + 17)$ using the definition of $g(x)$ .
$g(x + 17) = (x + 17) - 17$
Simplify the expression:
$(x + 17) - 17 = x$
Thus, $(g \circ f)(x) = x$ .

STEP 8

The domain of $(g \circ f)(x) = x$ is the set of all real numbers because there are no restrictions on $x$ .
The compositions are: $(f \circ g)(x) = x$ $(g \circ f)(x) = x$
The domain of each composition is all real numbers.

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QuestionFind $(f \circ g)(x)$ and $(g \circ f)(x)$ and the domain of each. $f(x)=x+17, g(x)=x-17$ $(f \circ g)(x)=$ $\square$ (Simplify your answer.)

Studdy Solution

STEP 1

1. We are given two functions: $f(x) = x + 17$ and $g(x) = x - 17$ .
2. We need to find the compositions $(f \circ g)(x)$ and $(g \circ f)(x)$ .
3. The domain of a composition of functions is the set of all $x$ such that the composition is defined.

STEP 2

1. Find the composition $(f \circ g)(x)$ .
2. Simplify $(f \circ g)(x)$ .
3. Find the domain of $(f \circ g)(x)$ .
4. Find the composition $(g \circ f)(x)$ .
5. Simplify $(g \circ f)(x)$ .
6. Find the domain of $(g \circ f)(x)$ .

STEP 3

To find $(f \circ g)(x)$ , we substitute $g(x)$ into $f(x)$ . This means we replace every $x$ in $f(x)$ with $g(x)$ .
$(f \circ g)(x) = f(g(x)) = f(x - 17)$

STEP 4

Now, simplify $f(x - 17)$ using the definition of $f(x)$ .
$f(x - 17) = (x - 17) + 17$
Simplify the expression:
$(x - 17) + 17 = x$
Thus, $(f \circ g)(x) = x$ .

STEP 5

The domain of $(f \circ g)(x) = x$ is the set of all real numbers because there are no restrictions on $x$ .

STEP 6

To find $(g \circ f)(x)$ , we substitute $f(x)$ into $g(x)$ . This means we replace every $x$ in $g(x)$ with $f(x)$ .
$(g \circ f)(x) = g(f(x)) = g(x + 17)$

STEP 7

Now, simplify $g(x + 17)$ using the definition of $g(x)$ .
$g(x + 17) = (x + 17) - 17$
Simplify the expression:
$(x + 17) - 17 = x$
Thus, $(g \circ f)(x) = x$ .

STEP 8

The domain of $(g \circ f)(x) = x$ is the set of all real numbers because there are no restrictions on $x$ .
The compositions are: $(f \circ g)(x) = x$ $(g \circ f)(x) = x$
The domain of each composition is all real numbers.

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QuestionFind (f∘g)(x)(f \circ g)(x)(f∘g)(x) and (g∘f)(x)(g \circ f)(x)(g∘f)(x) and the domain of each. f(x)=x+17,g(x)=x−17f(x)=x+17, g(x)=x-17f(x)=x+17,g(x)=x−17 (f∘g)(x)=(f \circ g)(x)=(f∘g)(x)= □\square□ (Simplify your answer.)

Studdy Solution

STEP 1

STEP 2

STEP 3

To find (f∘g)(x) (f \circ g)(x) (f∘g)(x), we substitute g(x) g(x) g(x) into f(x) f(x) f(x). This means we replace every x x x in f(x) f(x) f(x) with g(x) g(x) g(x).(f∘g)(x)=f(g(x))=f(x−17) (f \circ g)(x) = f(g(x)) = f(x - 17) (f∘g)(x)=f(g(x))=f(x−17)

STEP 4

Now, simplify f(x−17) f(x - 17) f(x−17) using the definition of f(x) f(x) f(x).f(x−17)=(x−17)+17 f(x - 17) = (x - 17) + 17 f(x−17)=(x−17)+17Simplify the expression:(x−17)+17=x (x - 17) + 17 = x (x−17)+17=xThus, (f∘g)(x)=x (f \circ g)(x) = x (f∘g)(x)=x.

STEP 5

The domain of (f∘g)(x)=x (f \circ g)(x) = x (f∘g)(x)=x is the set of all real numbers because there are no restrictions on x x x.

STEP 6

To find (g∘f)(x) (g \circ f)(x) (g∘f)(x), we substitute f(x) f(x) f(x) into g(x) g(x) g(x). This means we replace every x x x in g(x) g(x) g(x) with f(x) f(x) f(x).(g∘f)(x)=g(f(x))=g(x+17) (g \circ f)(x) = g(f(x)) = g(x + 17) (g∘f)(x)=g(f(x))=g(x+17)

STEP 7

Now, simplify g(x+17) g(x + 17) g(x+17) using the definition of g(x) g(x) g(x).g(x+17)=(x+17)−17 g(x + 17) = (x + 17) - 17 g(x+17)=(x+17)−17Simplify the expression:(x+17)−17=x (x + 17) - 17 = x (x+17)−17=xThus, (g∘f)(x)=x (g \circ f)(x) = x (g∘f)(x)=x.

STEP 8

The domain of (g∘f)(x)=x (g \circ f)(x) = x (g∘f)(x)=x is the set of all real numbers because there are no restrictions on x x x. The compositions are: (f∘g)(x)=x (f \circ g)(x) = x (f∘g)(x)=x (g∘f)(x)=x (g \circ f)(x) = x (g∘f)(x)=xThe domain of each composition is all real numbers.

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QuestionFind $(f \circ g)(x)$ and $(g \circ f)(x)$ and the domain of each. $f(x)=x+17, g(x)=x-17$ $(f \circ g)(x)=$ $\square$ (Simplify your answer.)

To find $(f \circ g)(x)$ , we substitute $g(x)$ into $f(x)$ . This means we replace every $x$ in $f(x)$ with $g(x)$ .
$(f \circ g)(x) = f(g(x)) = f(x - 17)$

Now, simplify $f(x - 17)$ using the definition of $f(x)$ .
$f(x - 17) = (x - 17) + 17$
Simplify the expression:
$(x - 17) + 17 = x$
Thus, $(f \circ g)(x) = x$ .

The domain of $(f \circ g)(x) = x$ is the set of all real numbers because there are no restrictions on $x$ .

To find $(g \circ f)(x)$ , we substitute $f(x)$ into $g(x)$ . This means we replace every $x$ in $g(x)$ with $f(x)$ .
$(g \circ f)(x) = g(f(x)) = g(x + 17)$

Now, simplify $g(x + 17)$ using the definition of $g(x)$ .
$g(x + 17) = (x + 17) - 17$
Simplify the expression:
$(x + 17) - 17 = x$
Thus, $(g \circ f)(x) = x$ .

The domain of $(g \circ f)(x) = x$ is the set of all real numbers because there are no restrictions on $x$ .
The compositions are: $(f \circ g)(x) = x$ $(g \circ f)(x) = x$
The domain of each composition is all real numbers.