Composite functions

 1.2 Composite Functions

 Definition Given two functions $$f(x)$$ and $$g(x)$$, the product of combination of two functions that written as $$fg(x)$$ or $$gf(x)$$ are defined by $$fg(x)=f[g(x)]$$ or $$gf(x)=g[f(x)]$$.

Properties of Composite Function
 Figure

 Description
• Function $$f$$ maps set $$P$$ to set $$Q$$, function $$g$$ maps set $$Q$$ to set $$R$$ and function $$gf$$ maps set $$P$$ to set $$R$$
• Given two functions $$f(x)$$ and $$g(x)$$, both functions can be combined and written as $$fg(x)$$ or $$gf(x)$$ which is defined as $$fg(x) = f[g(x)]$$ or $$gf(x)=g[f(x)]$$.
• In general, $$fg \neq gf$$$$f^2=ff$$$$f^3=fff$$, and so on.

Example
 Question

Given two functions $$f(x)=2x$$ and $$g(x)=x^2-5$$.

Determine the following composite functions.

(a) $$fg$$
(b) $$gf$$
(c) $$f^2$$
(d) $$g^2$$

 Solution

(a)

\begin{aligned} fg(x)&=f[g(x)] \\ &=f(x^2-5) \\ &=2(x^2-5) \\ &=2x^2-10 \end{aligned}

$$\therefore fg(x)=2x^2-10$$

(b)

\begin{aligned} gf(x)&=g[f(x)] \\ &=g(2x) \\ &=(2x)^2-5 \\ &=4x^2-5 \end{aligned}

$$\therefore gf(x)=4x^2-5$$

(c)

\begin{aligned} f^2(x)&=f[f(x)] \\ &=f(2x) \\ &=2(2x) \\ &=4x \end{aligned}

$$\therefore f^2(x)=4x$$

(d)

\begin{aligned} g^2(x)&=g[g(x)] \\ &=g(x^2-5) \\ &=(x^2-5)^2-5 \\ &=x^4-10x^2+25-5\\ &=x^4-10x^2+20 \end{aligned}

$$\therefore g^2(x)=x^4-10x^2+25$$

Example $$2$$
 Question

If $$f(x)=x-1$$ and $$g(x)=x^2-3x+4$$, find

(a) $$fg(2)$$,
(b) the values of $$x$$ when $$fg(x)=7$$
.

 Solution

(a)

\begin{aligned} fg(x)&=f[g(x)] \\ &=f(x^2-3x+4) \\ &=x^2-3x+4-1 \\ &=x^2-3x+3 \end{aligned}

Thus,

\begin{aligned} fg(2)&=(2)^2-3(2)+3 \\ &=1 \end{aligned}

(b)

\begin{aligned} fg(x)&=7 \\ x^2-3x+3&=7 \\ x^2-3x-4&=0 \\ (x+1)(x-4)&=0 \end{aligned}

Thus,

$$\therefore x=-1,\quad x=4$$

Example $$3$$
 Question

Given function $$f(x)=x-2$$. Find the function $$g(x)$$ if $$fg(x)=8x-7$$.

 Solution

\begin{aligned} f[g(x)]&=8x-7 \\ g(x)-2&=8x-7 \\ g(x)&=8x-7+2 \\ \end{aligned}

$$\therefore g(x)=8x-5$$

Composite functions

 1.2 Composite Functions

 Definition Given two functions $$f(x)$$ and $$g(x)$$, the product of combination of two functions that written as $$fg(x)$$ or $$gf(x)$$ are defined by $$fg(x)=f[g(x)]$$ or $$gf(x)=g[f(x)]$$.

Properties of Composite Function
 Figure

 Description
• Function $$f$$ maps set $$P$$ to set $$Q$$, function $$g$$ maps set $$Q$$ to set $$R$$ and function $$gf$$ maps set $$P$$ to set $$R$$
• Given two functions $$f(x)$$ and $$g(x)$$, both functions can be combined and written as $$fg(x)$$ or $$gf(x)$$ which is defined as $$fg(x) = f[g(x)]$$ or $$gf(x)=g[f(x)]$$.
• In general, $$fg \neq gf$$$$f^2=ff$$$$f^3=fff$$, and so on.

Example
 Question

Given two functions $$f(x)=2x$$ and $$g(x)=x^2-5$$.

Determine the following composite functions.

(a) $$fg$$
(b) $$gf$$
(c) $$f^2$$
(d) $$g^2$$

 Solution

(a)

\begin{aligned} fg(x)&=f[g(x)] \\ &=f(x^2-5) \\ &=2(x^2-5) \\ &=2x^2-10 \end{aligned}

$$\therefore fg(x)=2x^2-10$$

(b)

\begin{aligned} gf(x)&=g[f(x)] \\ &=g(2x) \\ &=(2x)^2-5 \\ &=4x^2-5 \end{aligned}

$$\therefore gf(x)=4x^2-5$$

(c)

\begin{aligned} f^2(x)&=f[f(x)] \\ &=f(2x) \\ &=2(2x) \\ &=4x \end{aligned}

$$\therefore f^2(x)=4x$$

(d)

\begin{aligned} g^2(x)&=g[g(x)] \\ &=g(x^2-5) \\ &=(x^2-5)^2-5 \\ &=x^4-10x^2+25-5\\ &=x^4-10x^2+20 \end{aligned}

$$\therefore g^2(x)=x^4-10x^2+25$$

Example $$2$$
 Question

If $$f(x)=x-1$$ and $$g(x)=x^2-3x+4$$, find

(a) $$fg(2)$$,
(b) the values of $$x$$ when $$fg(x)=7$$
.

 Solution

(a)

\begin{aligned} fg(x)&=f[g(x)] \\ &=f(x^2-3x+4) \\ &=x^2-3x+4-1 \\ &=x^2-3x+3 \end{aligned}

Thus,

\begin{aligned} fg(2)&=(2)^2-3(2)+3 \\ &=1 \end{aligned}

(b)

\begin{aligned} fg(x)&=7 \\ x^2-3x+3&=7 \\ x^2-3x-4&=0 \\ (x+1)(x-4)&=0 \end{aligned}

Thus,

$$\therefore x=-1,\quad x=4$$

Example $$3$$
 Question

Given function $$f(x)=x-2$$. Find the function $$g(x)$$ if $$fg(x)=8x-7$$.

 Solution

\begin{aligned} f[g(x)]&=8x-7 \\ g(x)-2&=8x-7 \\ g(x)&=8x-7+2 \\ \end{aligned}

$$\therefore g(x)=8x-5$$