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 $f,g$ 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:

   $f\left(g\right)=sing\text{and}g\left(x\right)=2x$

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 $f\left(g\right)$ .

Task!

Specify the functions $f,g$ for the composite functions  

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

Answer

Answer

Answer