03 1composite

Composite Functions

A composite function is formed when one function is put “inside” another.

• Each x value of the outside function is substituted with the inside function.

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Example 1 .

Notation

… … $f \left( g(x) \right)$ is read as f of g and can be written as $f_og(x)$

… … Similarly,

… … $g \left( f(x) \right)$ is read as g of f and can be written as $g_of(x)$

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Domain and Range of Composite Functions

The composite function $f \left( g(x) \right)$ is only defined if

• the range of g(x) is equal to or a subset of the domain of f(x)
• ie. The output from g(x) has to become the input to f(x) so it has to ‘fit’.

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The domain of $f\left( g(x) \right)$ is the domain of g(x)

The range of $f\left( g(x) \right)$ will be the output from f(x) given the input is the values coming out of g(x) (ie the range of g(x))

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Example 2

… … If $f(x) = \sqrt{x} \qquad \text{Domain: } x \in [0, \; \infty)$

… … And $g(x) = x^3 \qquad \text{Domain: } x \in R$

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… … Then $f \big( g(x) \big) = \sqrt{ x^3 }$

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… … Notice that the range of g(x) is $y \in R$

… … The range of g(x) is not a subset of the domain of f(x) (it doesn't fit)

… … So $f \big( g(x) \big)$ is undefined.

BUT

We could restrict the domain of g(x) so that the range of g(x) does fit into the domain of f(x)

… … ie we want the range of g(x) to be: $y \in [0, \; \infty)$ {or smaller}

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Maximal Domain

The maximal domain is the largest possible domain for g(x), which gives a range of g(x) which does fit into the domain of f(x).

• Sometimes the maximal domain is referred to as the implied domain

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Example 2 (continued)

… … In the example above,
… … the largest possible domain for g(x) (ie the maximal domain)
… … for which $\text{range } g(x) \subset \text{ domain } f(x)$ is:

… … $g(x) = x^3 \qquad \text{Domain: } x \in [0, \; \infty)$

… … Now the range of g(x) is $y \in [0, \; \infty)$

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… … With this restricted domain, $f \big( g(x) \big)$ is defined.

… … $f \big( g(x) \big) = \sqrt{ x^3 } \qquad x \in [0, \; \infty)$

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… … Remember, the domain of $f \big(g(x) \big)$ is the same as the domain of g(x).

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Example 3

… … If $f(x) = \sin(x) \qquad \qquad \text{domain: } x \in R \quad \text{ and range: } y \in [-1, \; 1]$

… … and $g(x) = \big(x + \frac{1}{2}\big)^2 \qquad \text{domain: } x \in R \quad \text{ and range: }y \in [0, \; \infty)$

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… … Then

… … $g \big( f(x) \big) = \big( \sin(x) + \frac{1}{2} \big)^2$

… … Domain of gof(x) is domain of f(x)$x \in R$

… … gof(x) is defined because range of f(x) “fits” into domain of g(x).

… … … $[-1, \; 1] \subset R$

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… … When [–1, 1] is used as the input for g(x), the output from g is $y \in \big[0, \; \frac{9}{4} \big]$

… … So range of gof(x) is $y \in \big[0, \; \frac{9}{4} \big]$

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It sometimes helps to map the transformation of domain into range

… … $x \longrightarrow f(x) \longrightarrow \qquad \quad \longrightarrow g(x) \quad \longrightarrow \quad y$

… … $R \rightarrow \sin(x) \rightarrow [-1, \; 1] \rightarrow \big(x + \frac{1}{2} \big)^2 \rightarrow \big[ 0, \; \frac{9}{4} \big]$

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Graphing Composite Functions

Graphing a composite function can be done using a variation on the Addition of Ordinates process.

Many composite graphs become complicated so graphing software is usually used.

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Example 4

… … Given $f(x) = \sin(x)$

… … and $g(x) = \big( x + \frac{1}{2} \big)^2$

… … Use technology to sketch $g \big( f(x) \big)$
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Solution

… … Domain: $x \in R$

… … Range: $y \in \big[0, \; \frac{9}{4} \big]$

… … Using CAS we get

… … y-intercept: $\big(0, \; \frac{1}{4} \big)$
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… … x-intercepts: $x \in \Big\{ 2\pi n - \dfrac{\pi}{6}, \; 2\pi n + \dfrac{7\pi}{6} \Big\} \quad \text{ for } n \in Z$

… … … ie $x \in \Big\{ \dots, \; -\dfrac{5\pi}{6}, \; -\dfrac{\pi}{6}, \; \dfrac{7\pi}{6}, \; \dfrac{11\pi}{6}, \; \dfrac{19\pi}{6}, \; \dots \Big\}$

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… … local max at $x \in \Big\{ n\pi + \dfrac{\pi}{2} \Big\} \quad \text{ for } n \in z$

… … … ie $x \in \Big\{ \dots, \; -\dfrac{\pi}{2}, \; \dfrac{\pi}{2}, \; \dfrac{3\pi}{2}, \; \dots \Big\}$

… … the local max alternates between $y = \frac{1}{4} \text{ and } y = \frac{9}{4}$

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