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### Composite Function Exercises Worked

Exercises from public domain textbooks. Solutions where provided do not necessary follow the same lines as the original.

#### X-1.

If {$h(x)=(f \circ g)(x)$} and {$h(x)=3\sqrt x+3$}, find {$g(x) \text{ if } f(x)=3\sqrt x+2$}

{\begin{align} h(x) &= f\left( g(x) \right) \cr 3\sqrt x+3 &= 3\sqrt{g(x)}+2 \cr 3\sqrt x+3 -2 &= 3\sqrt{g(x)} \cr \sqrt x+ {1 \over 3} &= \sqrt{g(x)} \cr \left( \sqrt x+ {1 \over 3} \right) ^ 2 &= g(x) \end{align}}

Check:

{\begin{align} h(x) &= f\left( g(x) \right)\cr &= 3 \sqrt{ \left( \sqrt x+ {1 \over 3} \right) ^ 2} + 2 \cr &= 3 \left( \sqrt x+ {1 \over 3} \right) + 2 \cr &= 3\sqrt x + (3){1 \over 3} + 2 \cr h(x) &= 3\sqrt x + 3 \end{align}}

Graphs:

Sources:

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Category: Math Precalculus Functions Calculus

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### December 25, 2018

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