QuestionFind and simplify: (a) $(f \circ g)(x)$ , (b) $(g \circ f)(x)$ , (c) $(f \circ f)(x)$ , (d) $(g \circ g)(x)$ , where $f(x)=x^{2}+8$ and $g(x)=x+6$ .

Studdy Solution

STEP 1

Assumptions1. The function $f(x)$ is given by $f(x)=x^{}+8$ . The function $g(x)$ is given by $g(x)=x+6$

STEP 2

The composition of two functions, say $f$ and $g$ , denoted by $(f \circ g)(x)$ , is defined as $f(g(x))$ . This means that we substitute the function $g(x)$ into the function $f(x)$ .
(a) To find $(f \circ g)(x)$ , we substitute $g(x)$ into $f(x)$ .
$(f \circ g)(x) = f(g(x))$

STEP 3

Substitute $g(x) = x+6$ into $f(x) = x^{2}+8$ .
$(f \circ g)(x) = (x+6)^{2}+8$

STEP 4

implify the expression.
$(f \circ g)(x) = x^{2} +12x +36 +8$

STEP 5

Further simplify the expression.
$(f \circ g)(x) = x^{2} +12x +44$

STEP 6

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

STEP 7

Substitute $f(x) = x^{2}+$ into $g(x) = x+6$ .
$(g \circ f)(x) = (x^{2}+) +6$

STEP 8

implify the expression.
$(g \circ f)(x) = x^{2} +14$

STEP 9

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

STEP 10

Substitute $f(x) = x^{2}+8$ into $f(x) = x^{2}+8$ .
$(f \circ f)(x) = (x^{2}+8)^{2}+8$

STEP 11

Expand and simplify the expression.
$(f \circ f)(x) = x^{4} +16x^{} +64 +8$

STEP 12

Further simplify the expression.
$(f \circ f)(x) = x^{4} +16x^{2} +72$

STEP 13

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

STEP 14

Substitute $g(x) = x+6$ into $g(x) = x+6$ .
$(g \circ g)(x) = (x+6) +6$

STEP 15

implify the expression.
$(g \circ g)(x) = x +12$ So, the solutions are(a) $(f \circ g)(x) = x^{2} +12x +44$ (b) $(g \circ f)(x) = x^{2} +14$ (c) $(f \circ f)(x) = x^{4} +x^{2} +72$ (d) $(g \circ g)(x) = x +12$

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QuestionFind and simplify: (a) $(f \circ g)(x)$ , (b) $(g \circ f)(x)$ , (c) $(f \circ f)(x)$ , (d) $(g \circ g)(x)$ , where $f(x)=x^{2}+8$ and $g(x)=x+6$ .

Studdy Solution

STEP 1

Assumptions1. The function $f(x)$ is given by $f(x)=x^{}+8$ . The function $g(x)$ is given by $g(x)=x+6$

STEP 2

The composition of two functions, say $f$ and $g$ , denoted by $(f \circ g)(x)$ , is defined as $f(g(x))$ . This means that we substitute the function $g(x)$ into the function $f(x)$ .
(a) To find $(f \circ g)(x)$ , we substitute $g(x)$ into $f(x)$ .
$(f \circ g)(x) = f(g(x))$

STEP 3

Substitute $g(x) = x+6$ into $f(x) = x^{2}+8$ .
$(f \circ g)(x) = (x+6)^{2}+8$

STEP 4

implify the expression.
$(f \circ g)(x) = x^{2} +12x +36 +8$

STEP 5

Further simplify the expression.
$(f \circ g)(x) = x^{2} +12x +44$

STEP 6

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

STEP 7

Substitute $f(x) = x^{2}+$ into $g(x) = x+6$ .
$(g \circ f)(x) = (x^{2}+) +6$

STEP 8

implify the expression.
$(g \circ f)(x) = x^{2} +14$

STEP 9

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

STEP 10

Substitute $f(x) = x^{2}+8$ into $f(x) = x^{2}+8$ .
$(f \circ f)(x) = (x^{2}+8)^{2}+8$

STEP 11

Expand and simplify the expression.
$(f \circ f)(x) = x^{4} +16x^{} +64 +8$

STEP 12

Further simplify the expression.
$(f \circ f)(x) = x^{4} +16x^{2} +72$

STEP 13

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

STEP 14

Substitute $g(x) = x+6$ into $g(x) = x+6$ .
$(g \circ g)(x) = (x+6) +6$

STEP 15

implify the expression.
$(g \circ g)(x) = x +12$ So, the solutions are(a) $(f \circ g)(x) = x^{2} +12x +44$ (b) $(g \circ f)(x) = x^{2} +14$ (c) $(f \circ f)(x) = x^{4} +x^{2} +72$ (d) $(g \circ g)(x) = x +12$

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QuestionFind and simplify: (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+8f(x)=x^{2}+8f(x)=x2+8 and g(x)=x+6g(x)=x+6g(x)=x+6.

Studdy Solution

STEP 1

Assumptions1. The function f(x)f(x)f(x) is given by f(x)=x+8f(x)=x^{}+8f(x)=x+8 . The function g(x)g(x)g(x) is given by g(x)=x+6g(x)=x+6g(x)=x+6

STEP 2

STEP 3

Substitute g(x)=x+6g(x) = x+6g(x)=x+6 into f(x)=x2+8f(x) = x^{2}+8f(x)=x2+8.(f∘g)(x)=(x+6)2+8 (f \circ g)(x) = (x+6)^{2}+8 (f∘g)(x)=(x+6)2+8

STEP 4

implify the expression.(f∘g)(x)=x2+12x+36+8 (f \circ g)(x) = x^{2} +12x +36 +8 (f∘g)(x)=x2+12x+36+8

STEP 5

Further simplify the expression.(f∘g)(x)=x2+12x+44 (f \circ g)(x) = x^{2} +12x +44 (f∘g)(x)=x2+12x+44

STEP 6

Similarly, 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).(g∘f)(x)=g(f(x)) (g \circ f)(x) = g(f(x)) (g∘f)(x)=g(f(x))

STEP 7

Substitute f(x)=x2+f(x) = x^{2}+f(x)=x2+ into g(x)=x+6g(x) = x+6g(x)=x+6.(g∘f)(x)=(x2+)+6 (g \circ f)(x) = (x^{2}+) +6 (g∘f)(x)=(x2+)+6

STEP 8

implify the expression.(g∘f)(x)=x2+14 (g \circ f)(x) = x^{2} +14 (g∘f)(x)=x2+14

STEP 9

To find (f∘f)(x)(f \circ f)(x)(f∘f)(x), we 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 10

Substitute f(x)=x2+8f(x) = x^{2}+8f(x)=x2+8 into f(x)=x2+8f(x) = x^{2}+8f(x)=x2+8.(f∘f)(x)=(x2+8)2+8 (f \circ f)(x) = (x^{2}+8)^{2}+8 (f∘f)(x)=(x2+8)2+8

STEP 11

Expand and simplify the expression.(f∘f)(x)=x4+16x+64+8 (f \circ f)(x) = x^{4} +16x^{} +64 +8 (f∘f)(x)=x4+16x+64+8

STEP 12

Further simplify the expression.(f∘f)(x)=x4+16x2+72 (f \circ f)(x) = x^{4} +16x^{2} +72 (f∘f)(x)=x4+16x2+72

STEP 13

To find (g∘g)(x)(g \circ g)(x)(g∘g)(x), we 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 14

Substitute g(x)=x+6g(x) = x+6g(x)=x+6 into g(x)=x+6g(x) = x+6g(x)=x+6.(g∘g)(x)=(x+6)+6 (g \circ g)(x) = (x+6) +6 (g∘g)(x)=(x+6)+6

STEP 15

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QuestionFind and simplify: (a) $(f \circ g)(x)$ , (b) $(g \circ f)(x)$ , (c) $(f \circ f)(x)$ , (d) $(g \circ g)(x)$ , where $f(x)=x^{2}+8$ and $g(x)=x+6$ .

Assumptions1. The function $f(x)$ is given by $f(x)=x^{}+8$ . The function $g(x)$ is given by $g(x)=x+6$

Substitute $g(x) = x+6$ into $f(x) = x^{2}+8$ .
$(f \circ g)(x) = (x+6)^{2}+8$

implify the expression.
$(f \circ g)(x) = x^{2} +12x +36 +8$

Further simplify the expression.
$(f \circ g)(x) = x^{2} +12x +44$

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

Substitute $f(x) = x^{2}+$ into $g(x) = x+6$ .
$(g \circ f)(x) = (x^{2}+) +6$

implify the expression.
$(g \circ f)(x) = x^{2} +14$

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

Substitute $f(x) = x^{2}+8$ into $f(x) = x^{2}+8$ .
$(f \circ f)(x) = (x^{2}+8)^{2}+8$

Expand and simplify the expression.
$(f \circ f)(x) = x^{4} +16x^{} +64 +8$

Further simplify the expression.
$(f \circ f)(x) = x^{4} +16x^{2} +72$

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

Substitute $g(x) = x+6$ into $g(x) = x+6$ .
$(g \circ g)(x) = (x+6) +6$