OP - Composition of Functions Lesson

Composition of Functions

The next thing to look at is composite functions.

Composition of functions is when we are given two functions, their composite (combined function) uses the output from one function as the input for the other function. We will see the common notation of f(x) and g(x), representing the two different functions.

Notation for composite functions is $LaTeX: f\circ g$ $f\circ g$ or f(g(x)).

The second notation allows us to see what the input is and what the output is.

In f(g(x)), the output for the function g(x) is used as the input for f(x).

In g(f(x)), the output for the function f(x) is used as the input for g(x).

The purpose of composing functions is often to evaluate the result for a specific number.

For example, if equation image indicator $LaTeX: f\left(x\right)=2x+3$ $f\left(x\right)=2x+3$ and $LaTeX: g\left(x\right)=x^2-x$ $g\left(x\right)=x^2-x$ , we might want to find $LaTeX: \left(f\:\circ\:g\right)\left(3\right)\:or\:\left(f\left(g\left(3\right)\right)\right)$ $\left(f\:\circ\:g\right)\left(3\right)\:or\:\left(f\left(g\left(3\right)\right)\right)$ .

To do this, we don't have to find equation image indicator $LaTeX: f\left(g\left(x\right)\right)$ $f\left(g\left(x\right)\right)$ . We can simply follow order of operation, doing what is in the ( ) first.

So we find g(3) by putting 3 in for x in the g(x) function.

equation image indicator $LaTeX: g\left(3\right)=3^2-3=9-3=6$ $g\left(3\right)=3^2-3=9-3=6$

Now we take g(3) which is 6 and put it in for x in the f(x) function.

equation image indicator $LaTeX: f\left(g\left(3\right)\right)=f\left(6\right)=2\left(6\right)+3=15$ $f\left(g\left(3\right)\right)=f\left(6\right)=2\left(6\right)+3=15$

Let's see how this is used.

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