Composite Functions (AQA A-Level Mathematics): Revision Notes

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2.8.2 Composite Functions

Composite Functions

A function is a 'thing' that 'eats' a number, does something to it, then 'spits out' the result.

f(x)=x2+2f(x) = x^2 + 2

Here, ff is the function which 'eats' a number, squares it, adds 22, then spits out the result.

Sometimes we can apply multiple functions to a quantity.

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e.g.

f(x)=exf(x) = e^xg(x)=x2−3g(x) = x^2 - 3fg(x)=f(g(x))=f(x2−3)=ex2−3gg(x)=g(g(x))=g(x2−3)=(x2−3)2−3\begin{align*} fg(x) & = f(g(x)) = f(x^2 - 3) = e^{x^2-3} \\ gg(x) & = g(g(x)) = g(x^2 - 3) = (x^2 - 3)^2 - 3 \end{align*}

Easier way of thinking about it:

f[□]=e□f \left[ \square \right] = e^{\square} g[□]=□2−3g \left[ \square \right] = \square^2 - 3 ff(x)=f(f(x))=f(ex)=eexggf(y2)=gg(ey2)=g((ey2)2−3)=((ey2)2−3))2−3\begin{align*} ff(x) & = f(f(x)) = f(e^x) = e^{e^x} \\ gg^{f(y^2)} &= gg^{(e^{y^2})} = g(^{(e^{y^2})^2}-3) = ({(e^{y^2})^2}-3))^2-3 \end{align*}

Numerical expressions can also be fed into a function.

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e.g.

fgf(2)=fg(f(2))=fg(e2)=f(e4−3)=ee4−3\begin{align*} fgf(2) & = fg(f(2)) = fg(e^2) \\ & = f(e^4 - 3) = e^{e^4-3} \end{align*}

Key point: The function closest to the bracket is the one that gets applied first.

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The functions ff and gg are defined for all real values of xx by

f[□]=3□−2g[□]=3□+7f \left[ \square \right] = 3\square-2 \\\quad g \left[ \square \right] = 3\square+7f(x)=3x−2andg(x)=3x+7f(x) = 3x - 2 \quad \text{and} \quad g(x) = 3x + 7

Find the exact coordinates of the point at which

(i) the graph of y=fg(x)y = fg(x) meets the xx-axis.

y=fg(x)=f(g(x))=f(3x+7)=3(3x+7)−2=9x+21−2=9x+19\begin{align*} y & = fg(x) \\ & = f(g(x)) \\ & = f(3x + 7) \\ & = 3(3x + 7) - 2 \\ & = 9x + 21 - 2 \\ & = 9x + 19 \end{align*}

Let y=0y = 0,

0=9x+199x=−19x=−199\begin{align*} 0 & = 9x + 19 \\ 9x & = -19 \\ x & = -\frac{19}{9} \end{align*}∴(−199,0)\therefore \quad \left( -\frac{19}{9}, 0 \right)

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