2.10.2 Composite & Inverse functions
A composite function is a function that is applied to the output of another function.
A composite function is denoted by the symbol \( fg(x) \).
- If you enter a number into \( fg(x) \)
- Enter the number into \( g(x) \)
- Input the result of \( g(x) \) into \( f(x) \)
Tip:
Check that you are using the functions in the correct order.
Worked example:
\( f(x)=7-2x \), \( g(x)=\frac{10}{x} \), \( x≠0 \), \( h(x)=27^x \)
Find
- \( f(-3) \)
Substitute \( x=-3 \) into the function \( f(x) \) to find \( f(-3) \)
\( f(-3)=7-2(-3) \)
\( =7+6 \)
\( =13 \)
\( f(-3)=13 \)
- \( hg(30) \)
To work out a composite function, start with the function closest to \( x \).
Start by substituting \( x=30 \) into \( g(x) \).
\( g(30)=\frac{10}{30}=\frac{1}{3} \)
Substitute the result into \( h(x) \) to work out \( hg(30) \)
\( hg(30)=h(\frac{1}{3}) \)
\( =27^{\frac{1}{3}} \)
\( =3 \)
\( hg(30)=3 \)
Inverse Functions:
An inverse function performs the exact opposite of the function from which it was derived.
If the function is denoted by \( f(x) \) then the inverse function is denoted by \( f^{-1}(x).
To find the inverse function
- In the function equation, replace \( f(x) \) with \( y \)
- Replace every \( x \) by a \( y \)
- Solve for \( y \)
- Replace \( y \) by \( f^{-1}(x) \)
Worked example:
Find \( f^{-1}(x) \) when \( f(x)=7x-2 \).
Replace \( f(x) \) with \( y \)
\( y=7x-2 \)
Solve for \( x \)
\( y+2=7x \)
\( x=\frac{y+2}{7} \)
Interchange \( x \) and \( y \)
\( y=\frac{x+2}{7} \)
Replace \( y \) by \( f^{-1}(x) \)
\( f^{-1}(x)=\frac{x+2}{7} \)
\( f^{-1}(x)=\frac{x+2}{7} \)
Test yourself:
Question 1:
\( f(x)=7-2x \), \( g(x)=\frac{10}{x} \), \( x≠0 \), \( h(x)=27^x \)
Simplify, giving your answer as a single fraction.
\( \frac{1}{f(x)}+g(x) \)
[3]
Question 2:
\( g(x)=8x-5 \), \( h(x)=x^2+6 \)
Show that \( hg(x)=19 \) simplifies to \( 16x^2-20x+3=0 \).
[3]
Use the quadratic formula to solve \( 16x^2-20x+3=0 \).
Show all your working and give your answers correct to \( 2 \) decimal places.
[4]
Question 3:
\( f(x)=8-3x \), \( g(x)=\frac{10}{x+1} \), \( x≠-1 \), \( h(x)=2^x \)
Find
- \( hf(\frac{8}{3}) \),
[2]
- \( gh(-2) \),
[2]
- \( g^{-1}(x) \),
[3]
- \( f^{-1}f(5) \).
[1]
Question 4:
\( f(x)=4x-1 \), \( g(x)=x^2 \), \( h(x)=3^{-x} \)
Show that \( g(3x-2)-h(-3) \) can be written as \( 9x^2-12x-23 \).
[2]
Find \( x \) when \( f(61)=h(x) \).
[2]
Question 5:
\( f(x)=7x-2 \), \( g(x)=x^2+1 \), \( h(x)=3^x \)
\( gg(x)=ax^4+bx^2+c \)
- Find the value of \( a \),\( b \) and \( c \).
[3]
- Find \( x \) when \( hf(x)=81 \).
[3]