Learning objectives
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), where the function g is applied to the result of applying the function f to x.
The output, or function values, of the function f\left(x\right) have become the input, or x-values, of the function g\left(x\right). We introduce a new symbol \circ to represent this new function.
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).
The domain of \left(g \circ f\right)\left(x\right) is restricted to all x-values in the domain of f whose range values, f\left(x\right), are in the domain of g.
Consider the following pair of functions:
\begin{aligned} f\left(x\right) & = -5x+5\\\ g\left(x\right) & = 2x^2+3x-10 \end{aligned}
Find \left(f \circ g\right)\left(x\right)
Find \left(g \circ f\right)\left(x\right)
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.
State the function for h\left(g\right), the height of the grain in the container, in terms of g.
State the function for g\left(t\right), the amount of grain in the tank after t seconds.
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.
Explain what A\left(t \right) represents.
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).
Consider the functions f\left(x\right)=x^2 and g\left(x\right)=x+5.
If y=f\left(g\left(x\right)\right), state the equation for y.
Graph the function y.
Describe the transformation of f\left(x\right) that y corresponds to.
Composite function is a function created when one function is substituted into another function.
Composition: \left(f \circ g\right)\left(x\right)= f\left(g\left(x\right)\right)
Function composition is not commutative.
That is: \left(f \circ g\right)\left(x\right) \neq \left(g \circ f\right)\left(x\right) in most cases.