Composition and Decomposition

Hint.

For any input \(x\) and functions \(f\) and \(g\text{,}\) this action defines a composite function, which we write as \(f \circ g\) such that

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\(\displaystyle{(f\circ g)(x) = f(g(x))}\)

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The domain of the composite function \(f\circ g\) is all \(x\) such that \(x\) is in the domain of \(g\) and \(g(x)\) is in the domain of \(f\text{.}\)

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Answer 1.

\(\sqrt{x^{2}+9}-1\)

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Answer 2.

\(\left(\sqrt{x}-1\right)^{2}+9\)

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3.

Suppose \(f(x) = x^{7}+6\) and \(g(x) = \sqrt{x}\text{.}\) Then

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\(f(g(x))\) =

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\(g(f(x))\) =

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Answer 1.

\(\left(\sqrt{x}\right)^{7}+6\)

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Answer 2.

\(\sqrt{x^{7}+6}\)

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4.

\(x\)

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2

3

4

5

6

7

8

9

\(f(x)\)

7

2

9

9

6

2

6

1

8

\(g(x)\)

4

3

6

6

8

5

8

5

9

Use the table to evaluate the expression,

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\((f\circ g)(6)=\)

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Answer.

\(6\)

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5.

\(x\)

1

2

3

4

5

6

7

8

9

\(f(x)\)

8

8

4

6

6

7

7

5

1

\(g(x)\)

9

8

6

5

9

2

7

1

3

Use the table to evaluate the expression,

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\((g\circ f)(4)=\)

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Answer.

\(2\)

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6.

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\(y = f(x)\)		\(y = g(x)\)

Use the graphs of \(y = f(x)\) and \(y = g(x)\) above to the find the function value.

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(a) \((g\circ f)(2) =\)

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(b) \((g\circ g)(1) =\)

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If the video does not work, click here to watch on YouTube.

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Answer 1.

\(1\)

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Answer 2.

\(2\)

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7.

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\(y = f(x)\)		\(y = g(x)\)

Use the graphs of \(y = f(x)\) and \(y = g(x)\) above to the find the function value.

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(a) \((g\circ f)(1) =\)

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(b) \((g\circ g)(2) =\)

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Answer 1.

\(2\)

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Answer 2.

\(3\)

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8.

Answer 1.

\(3\mathopen{}\left(-2x+2\right)^{2}+-2\mathopen{}\left(-2x+2\right)+4\)

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Answer 2.

\(-2\mathopen{}\left(3x^{2}+-2x+4\right)+2\)

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Answer 3.

\(\left(-\infty ,\infty \right)\)

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Answer 4.

\(\left(-\infty ,\infty \right)\)

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9.

Express the function \(y = \sqrt{ x^2 + 3 }\) as a composition \(y = f(g(x))\) of two simpler functions \(y = f(u)\) and \(u = g(x)\text{.}\)

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\(f(u)\) =

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\(g(x)\) =

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Answer 1.

sqrt(u)

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Answer 2.

x^2+3

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10.

Express the function \(y = 4 (x - 2)^{3}\) as a composition \(y = f(g(x))\) of two simpler functions \(y = f(u)\) and \(u = g(x)\text{.}\)

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\(f(u)\) =

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\(g(x)\) =

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Answer 1.

4*u^3

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Answer 2.

x-2

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11.

Express the function \(y = 5 x^{8} - 5\) as a composition \(y = f(g(x))\) of two simpler functions \(y = f(u)\) and \(u = g(x)\text{.}\)

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\(f(u)\) =

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\(g(x)\) =

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Answer 1.

5*u-5

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Answer 2.

x^8

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12.

If \(f(g(x)) = 6(x^{5}+4)^{7}\) and \(g(x) = x^{5}+4\text{,}\) find \(f(x)\text{.}\)

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\(f(x)\) =

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Answer.

\(6x^{7}\)

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Worksheet Composition and Decomposition

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Worksheet Composition and Decomposition

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Worksheet Composition and Decomposition