Composite functions

Example: Given, $f(x)=7x-2$ and $g(x)=11x+3$, find $(f+g)(x)$, $(f-g)(x)$, $(f\cdot g)(x)$ and $\left(\dfrac{f}{g}\right)(x)$.

Solution:

$(f+g)(x)=(7x-2)+(11x+3)$

Combining like terms,

$(f+g)(x)=18x+1$

$(f-g)(x)=(7x-2)-(11x+3)$

Combining like terms,

$(f-g)(x)=-4x-5$

$(f\cdot g)(x)=(7x-2)\cdot(11x+3)$

Multiplying and combining similar terms,

$(f\cdot g)(x)=77x^2-x-6$

$\left(\dfrac{f}{g}\right)(x)=\dfrac{7x-2}{11x+3}$

Composition of functions:

In general, a function is written in terms of a variable. That is, $f(x)$ is a function of the variable $x$, where $x$ is called the argument of the function. Sometimes the argument is itself another function, as in $f(g(x))$, where $g(x)$ is another function. We call $f(g(x))$ the composition of $f$ and $g$. To denote a composition of functions, we use a small circle between the functions.

That is, the composition of $f$ and $g$ is denoted $f\circ g$ and is defined as

$f\circ g=f(g(x))$

Example 1: Given $f(x)=x^2-7$ and $g(x)=\sqrt{x+10}$. Find $f\circ g$ and $g\circ f$.

Solution:

$(f\circ g)(x) =f(g(x))$

Take the function $f$ and replace $x$ with $g(x)$, that is, $\sqrt{x+10}$:

$(f\circ g)(x) =(\sqrt{x+10})^2-7$

$\hphantom{(f\circ g)(x)}=x+10-7$

$\hphantom{(f\circ g)(x)}=x+3$

$(g\circ f)(x) =g(f(x))$

Take the function $g$ and replace $x$ with $f(x)$, that is, $x^2-7$:

$(g\circ f)(x) =\sqrt{x^2-7+10}$

$\hphantom{(g\circ f)(x)}=\sqrt{x^2+3}$

Note that $f\circ g$ and $g\circ f$ are not equal.

Example 2: Given $f(x)=x^2+5$ and $g(x)=\sqrt{2x-8}$. Find $(f\circ g) (6)$ and $(g\circ f)(6)$.

Solution:

$(f\circ g)(6)=f(g(6))$

First find $g(6)$,

$g(6)=\sqrt{2\cdot 6 -8}$

$\hphantom{g(6)}=\sqrt{12-8}$

$\hphantom{g(6)}=\sqrt{4}$

$\hphantom{g(6)}=2$

Now, replace $x$ in $f$ with this value,

$(f\circ g)(6)=f(2)$

$\hphantom{(f\circ g)(6)}=2^2+5$

$\hphantom{(f\circ g)(6)}=9$

$(g\circ f)(6)=g(f(6))$

First find $f(6)$,

$f(6)=6^2+5$

$\hphantom{f(6)}=36+5$

$\hphantom{f(6)}=41$

Now, replace $x$ in $g$ with this value,

$(g\circ f)(6)=g(41)$

$\hphantom{(g\circ f)(6)}=\sqrt{2\cdot 41-8}$

$\hphantom{(g\circ f)(6)}=\sqrt{82-8}$

$\hphantom{(g\circ f)(6)}=\sqrt{74}$