QuestionFind the compositions of the functions: (a) $(f \circ g)(x)$ , (b) $(g \circ f)(x)$ , (c) $(f \circ f)(x)$ , (d) $(g \circ g)(x)$ , where $f(x)=x^{2}-4 x$ and $g(x)=x-4$ .

Studdy Solution

STEP 1

Assumptions1. The function $f(x)$ is given by $f(x)=x^{}-4 x$ . The function $g(x)$ is given by $g(x)=x-4$
3. The composition of functions is defined as $(f \circ g)(x) = f(g(x))$ and $(g \circ f)(x) = g(f(x))$

STEP 2

To find $(f \circ g)(x)$ , we need to substitute $g(x)$ into $f(x)$ .
$(f \circ g)(x) = f(g(x))$

STEP 3

Substitute $g(x)$ into $f(x)$ .
$(f \circ g)(x) = (g(x))^{2} -g(x)$

STEP 4

Substitute $x-4$ for $g(x)$ .
$(f \circ g)(x) = (x-4)^{2} -4(x-4)$

STEP 5

Expand the expression.
$(f \circ g)(x) = x^{2} -8x +16 -4x +16$

STEP 6

implify the expression.
$(f \circ g)(x) = x^{2} -12x +32$

STEP 7

To find $(g \circ f)(x)$ , we need to substitute $f(x)$ into $g(x)$ .
$(g \circ f)(x) = g(f(x))$

STEP 8

Substitute $f(x)$ into $g(x)$ .
$(g \circ f)(x) = f(x) -4$

STEP 9

Substitute $x^{2}-4x$ for $f(x)$ .
$(g \circ f)(x) = x^{2} -4x -4$

STEP 10

To find $(f \circ f)(x)$ , we need to substitute $f(x)$ into $f(x)$ .
$(f \circ f)(x) = f(f(x))$

STEP 11

Substitute $f(x)$ into $f(x)$ .
$(f \circ f)(x) = (f(x))^{} -4f(x)$

STEP 12

Substitute $x^{2}-4x$ for $f(x)$ .
$(f \circ f)(x) = (x^{2}-4x)^{2} -4(x^{2}-4x)$

STEP 13

Expand and simplify the expression.
$(f \circ f)(x) = x^{} -16x^{3} +64x^{2} -x^{2} +16x$

STEP 14

Combine like terms.
$(f \circ f)(x) = x^{4} -16x^{3} +60x^{2} +16x$

STEP 15

To find $(g \circ g)(x)$ , we need to substitute $g(x)$ into $g(x)$ .
$(g \circ g)(x) = g(g(x))$

STEP 16

Substitute $g(x)$ into $g(x)$ .
$(g \circ g)(x) = g(x) -4$

STEP 17

Substitute $x-4$ for $g(x)$ .
$(g \circ g)(x) = x -4 -4$

STEP 18

implify the expression.
$(g \circ g)(x) = x -8$ So, the solutions are(a) $(f \circ g)(x) = x^{2} -12x +32$ (b) $(g \circ f)(x) = x^{2} -4x -4$ (c) $(f \circ f)(x) = x^{4} -16x^{3} +60x^{2} +16x$ (d) $(g \circ g)(x) = x -8$

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QuestionFind the compositions of the functions: (a) (f∘g)(x)(f \circ g)(x)(f∘g)(x), (b) (g∘f)(x)(g \circ f)(x)(g∘f)(x), (c) (f∘f)(x)(f \circ f)(x)(f∘f)(x), (d) (g∘g)(x)(g \circ g)(x)(g∘g)(x), where f(x)=x2−4xf(x)=x^{2}-4 xf(x)=x2−4x and g(x)=x−4g(x)=x-4g(x)=x−4.

Studdy Solution

STEP 1

STEP 2

To find (f∘g)(x)(f \circ g)(x)(f∘g)(x), we need to substitute g(x)g(x)g(x) into f(x)f(x)f(x).(f∘g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f∘g)(x)=f(g(x))

STEP 3

Substitute g(x)g(x)g(x) into f(x)f(x)f(x).(f∘g)(x)=(g(x))2−g(x)(f \circ g)(x) = (g(x))^{2} -g(x)(f∘g)(x)=(g(x))2−g(x)

STEP 4

Substitute x−4x-4x−4 for g(x)g(x)g(x).(f∘g)(x)=(x−4)2−4(x−4)(f \circ g)(x) = (x-4)^{2} -4(x-4)(f∘g)(x)=(x−4)2−4(x−4)

STEP 5

Expand the expression.(f∘g)(x)=x2−8x+16−4x+16(f \circ g)(x) = x^{2} -8x +16 -4x +16(f∘g)(x)=x2−8x+16−4x+16

STEP 6

implify the expression.(f∘g)(x)=x2−12x+32(f \circ g)(x) = x^{2} -12x +32(f∘g)(x)=x2−12x+32

STEP 7

To find (g∘f)(x)(g \circ f)(x)(g∘f)(x), we need to substitute f(x)f(x)f(x) into g(x)g(x)g(x).(g∘f)(x)=g(f(x))(g \circ f)(x) = g(f(x))(g∘f)(x)=g(f(x))

STEP 8

Substitute f(x)f(x)f(x) into g(x)g(x)g(x).(g∘f)(x)=f(x)−4(g \circ f)(x) = f(x) -4(g∘f)(x)=f(x)−4

STEP 9

Substitute x2−4xx^{2}-4xx2−4x for f(x)f(x)f(x).(g∘f)(x)=x2−4x−4(g \circ f)(x) = x^{2} -4x -4(g∘f)(x)=x2−4x−4

STEP 10

To find (f∘f)(x)(f \circ f)(x)(f∘f)(x), we need to substitute f(x)f(x)f(x) into f(x)f(x)f(x).(f∘f)(x)=f(f(x))(f \circ f)(x) = f(f(x))(f∘f)(x)=f(f(x))

STEP 11

Substitute f(x)f(x)f(x) into f(x)f(x)f(x).(f∘f)(x)=(f(x))−4f(x)(f \circ f)(x) = (f(x))^{} -4f(x)(f∘f)(x)=(f(x))−4f(x)

STEP 12

Substitute x2−4xx^{2}-4xx2−4x for f(x)f(x)f(x).(f∘f)(x)=(x2−4x)2−4(x2−4x)(f \circ f)(x) = (x^{2}-4x)^{2} -4(x^{2}-4x)(f∘f)(x)=(x2−4x)2−4(x2−4x)

STEP 13

Expand and simplify the expression.(f∘f)(x)=x−16x3+64x2−x2+16x(f \circ f)(x) = x^{} -16x^{3} +64x^{2} -x^{2} +16x(f∘f)(x)=x−16x3+64x2−x2+16x

STEP 14

Combine like terms.(f∘f)(x)=x4−16x3+60x2+16x(f \circ f)(x) = x^{4} -16x^{3} +60x^{2} +16x(f∘f)(x)=x4−16x3+60x2+16x

STEP 15

To find (g∘g)(x)(g \circ g)(x)(g∘g)(x), we need to substitute g(x)g(x)g(x) into g(x)g(x)g(x).(g∘g)(x)=g(g(x))(g \circ g)(x) = g(g(x))(g∘g)(x)=g(g(x))

STEP 16

Substitute g(x)g(x)g(x) into g(x)g(x)g(x).(g∘g)(x)=g(x)−4(g \circ g)(x) = g(x) -4(g∘g)(x)=g(x)−4

STEP 17

Substitute x−4x-4x−4 for g(x)g(x)g(x).(g∘g)(x)=x−4−4(g \circ g)(x) = x -4 -4(g∘g)(x)=x−4−4

STEP 18

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QuestionFind the compositions of the functions: (a) $(f \circ g)(x)$ , (b) $(g \circ f)(x)$ , (c) $(f \circ f)(x)$ , (d) $(g \circ g)(x)$ , where $f(x)=x^{2}-4 x$ and $g(x)=x-4$ .

To find $(f \circ g)(x)$ , we need to substitute $g(x)$ into $f(x)$ .
$(f \circ g)(x) = f(g(x))$

Substitute $g(x)$ into $f(x)$ .
$(f \circ g)(x) = (g(x))^{2} -g(x)$

Substitute $x-4$ for $g(x)$ .
$(f \circ g)(x) = (x-4)^{2} -4(x-4)$

Expand the expression.
$(f \circ g)(x) = x^{2} -8x +16 -4x +16$

implify the expression.
$(f \circ g)(x) = x^{2} -12x +32$

To find $(g \circ f)(x)$ , we need to substitute $f(x)$ into $g(x)$ .
$(g \circ f)(x) = g(f(x))$

Substitute $f(x)$ into $g(x)$ .
$(g \circ f)(x) = f(x) -4$

Substitute $x^{2}-4x$ for $f(x)$ .
$(g \circ f)(x) = x^{2} -4x -4$

To find $(f \circ f)(x)$ , we need to substitute $f(x)$ into $f(x)$ .
$(f \circ f)(x) = f(f(x))$

Substitute $f(x)$ into $f(x)$ .
$(f \circ f)(x) = (f(x))^{} -4f(x)$

Substitute $x^{2}-4x$ for $f(x)$ .
$(f \circ f)(x) = (x^{2}-4x)^{2} -4(x^{2}-4x)$

Expand and simplify the expression.
$(f \circ f)(x) = x^{} -16x^{3} +64x^{2} -x^{2} +16x$

Combine like terms.
$(f \circ f)(x) = x^{4} -16x^{3} +60x^{2} +16x$

To find $(g \circ g)(x)$ , we need to substitute $g(x)$ into $g(x)$ .
$(g \circ g)(x) = g(g(x))$

Substitute $g(x)$ into $g(x)$ .
$(g \circ g)(x) = g(x) -4$

Substitute $x-4$ for $g(x)$ .
$(g \circ g)(x) = x -4 -4$