• 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.

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) \(g^2\)

(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)

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

\(\therefore g^2(x)=x^4-10x^2+20.\)

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

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

(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,

\(\therefore x=-1,\quad x=4.\)

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

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

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

• 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.

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) \(g^2\)

(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)

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

\(\therefore g^2(x)=x^4-10x^2+20.\)

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

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

(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,

\(\therefore x=-1,\quad x=4.\)

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

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

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