### 1 The meaning of a function of a function

When we use a function like $sin2x$ or ${e}^{lnx}$ or $\sqrt{{x}^{2}+1}$ we are in fact dealing with a composite function or function of a function .

$sin2x$ is the sine function of $2x$ . This is, in fact, how we ‘read’ it:

$sin2x$ is read ‘sine of $2x$

Similarly ${e}^{lnx}$ is the exponential function of the logarithm of $x$ :

${e}^{lnx}$ is read ‘ $e$ to the power of $lnx$

Finally $\sqrt{{x}^{2}+1}$ is also a composite function. It is the square root function of the polynomial ${x}^{2}+1$ :

$\sqrt{{x}^{2}+1}$ is read as the ‘square root of $\left({x}^{2}+1\right)$

When we talk about a function of a function in a general setting we will use the notation $f\left(g\left(x\right)\right)$ where both $f$ and $g$ are functions.

##### Example 11

Specify the functions $f,\phantom{\rule{1em}{0ex}}g$ for the composite functions

1. $sin2x$
2. $\sqrt{{x}^{2}+1}$
3. ${e}^{lnx}$
##### Solution
1. Here $f$ is the sine function and $g$ is the polynomial $2x$ . We often write:

$\phantom{\rule{2em}{0ex}}f\left(g\right)=sing\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}g\left(x\right)=2x$

2. Here $f\left(g\right)=\sqrt{g}$ and $g\left(x\right)={x}^{2}+1$
3. Here $f\left(g\right)={e}^{g}$ and $g\left(x\right)=lnx$

In each case the original function of $x$ is obtained when $g\left(x\right)$ is substituted into $f\left(g\right)$ .

Specify the functions $f,\phantom{\rule{1em}{0ex}}g$ for the composite functions

1. $cos\left(3{x}^{2}-1\right)$
2. $sinh\left({e}^{x}\right)$
3. ${\left({x}^{2}+3x-1\right)}^{1∕3}$

$f\left(g\right)=cosg\phantom{\rule{2em}{0ex}}g\left(x\right)=3{x}^{2}-1$

$f\left(g\right)=sinhg\phantom{\rule{2em}{0ex}}g\left(x\right)={e}^{x}$

$f\left(g\right)={g}^{1∕3}\phantom{\rule{2em}{0ex}}g\left(x\right)={x}^{2}+3x-1$