We can create a **composite function** using an operation that combines two functions f and g and produces a function h such that h\left(x\right)=g\left(f\left(x\right)\right). The output, or function values, of the function f\left(x\right) become the input, or x-values, of the function g\left(x\right).

Let f(x)= 2x+5 and g(x)= x-1

The composition of f with g is:

\displaystyle f(g(x)) | \displaystyle = | \displaystyle 2(x-1)+5 |

\displaystyle = | \displaystyle 2x-2+5 | |

\displaystyle = | \displaystyle 2x+3 |

Find the composition of g with f.

g(f(x))= â¬š

Is g(f(x)) the same as f(g(x))? Explain.

Evaluate g(2). Then evaluate f(g(2)).

g(2)= â¬š

f(g(2))=â¬š

- Explain what f(g(2)) represents.

The symbol \circ can also be used to represent a composite function. f\left(g\left(x\right)\right)=\left(f\circ g\right)\left(x\right)

In a composition of functions, the inner function is evaluated first, followed by the outer function. For example, in the composition g\left(f\left(x\right)\right), the function f is applied first, followed by the function g. This means that \left(g \circ f\right)\left(x\right) is not necessarily equal to \left(f \circ g\right)\left(x\right).

We can use graphs to evaluate the composition of functions. For example, consider the graphs of f(x) and g(x) shown. We will use these graphs to find g(f(8)).

For f(x)= x^2 + 5 and g(x)=3x-2:

a

Find f(-1).

Worked Solution

b

Find g\left(f(-1)\right).

Worked Solution

Consider the graphs of f(x) and g(x) shown.

a

Find g(-3).

Worked Solution

b

Find f(g(-3)).

Worked Solution

Consider the following pair of functions:

\begin{aligned} f\left(x\right) & = -5x+5\\\ g\left(x\right) & = 2x^2+3x-10 \end{aligned}

a

Find \left(f \circ g\right)\left(x\right)

Worked Solution

b

Find \left(g\circ f\right)\left(x\right)

Worked Solution

c

Does \left(f\circ g\right)\left(x\right)=\left(g\circ f\right)\left(x\right)?

Worked Solution

d

Compare the domain and range of \left(f\circ g\right)\left(x\right) and \left(g\circ f\right)\left(x\right).

Worked Solution

Use the graphs of g\left(x\right) and f\left(x\right) to find (f\circ g)\left(1\right)

Worked Solution

A cylindrical tank initially contains 200 \text{ in}^3 of grain and starts being filled at a constant rate of 40 \text{ in}^3 per second.

The radius of the tank is 12 inches. Let g be the amount of grain in the container after t seconds.

a

State the function for h\left(g\right), the height of the grain in the container, in terms of g.

Worked Solution

b

State the function for g\left(t\right), the amount of grain in the tank after t seconds.

Worked Solution

c

The function A\left(t\right) is defined as A\left(t\right)=\left( h \circ g \right)\left(t\right). Form an equation for A\left(t\right) in terms of t.

Worked Solution

d

Explain what A\left(t \right) represents.

Worked Solution

e

If the barrel can hold 10\,000 \text{ in}^3 of grain, determine the domains of g\left(t\right), h\left(g\right) and A\left(t\right).

Worked Solution

Idea summary

In a composition of functions, the inner function is evaluated first, followed by the outer function.\left(f \circ g\right)\left(x\right)= f\left(g\left(x\right)\right)

\left(g \circ f\right)\left(x\right)= g\left(f\left(x\right)\right)