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
 $sin2x$
 $\sqrt{{x}^{2}+1}$
 ${e}^{lnx}$
Solution

Here
$f$
is the sine function and
$g$
is the polynomial
$2x$
. We often write:
$\phantom{\rule{2em}{0ex}}f\left(g\right)=sing\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}g\left(x\right)=2x$
 Here $f\left(g\right)=\sqrt{g}$ and $g\left(x\right)={x}^{2}+1$
 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 $f\left(g\right)$ .
Task!
Specify the functions $f,\phantom{\rule{1em}{0ex}}g$ for the composite functions
 $cos\left(3{x}^{2}1\right)$
 $sinh\left({e}^{x}\right)$
 ${\left({x}^{2}+3x1\right)}^{1\u22153}$
$f\left(g\right)=cosg\phantom{\rule{2em}{0ex}}g\left(x\right)=3{x}^{2}1$
$f\left(g\right)=sinhg\phantom{\rule{2em}{0ex}}g\left(x\right)={e}^{x}$
$f\left(g\right)={g}^{1\u22153}\phantom{\rule{2em}{0ex}}g\left(x\right)={x}^{2}+3x1$