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Functions can be combined in another more subtle way. A composition of functions is when the output of one function is plugged into the input of another function. This is written as

This expression is pronounced as “f of g of x.” The expression on the left for the composition of functions is most similar to what you would type into a computer program. When the range of g(x) is within the domain of f(x) the composition of functions can be calculated for the original input x.

Example 1.Calculate f(g(3)) given the two functions

Solution 1. We can write an expression for the composition of functions.

Since the exponential is defined everywhere and polynomials are defined everywhere we don’t have any conflicts with the domain.

Repeated composition of functions can be calculated such as f(f(x)) or f(f(f(x))). If we wanted to do this 100 times we would have a fairly large expression so there is a shorthand notation of $latex f^n(x)&s=1$ for n repeated compositions. If $latex f(x) = (x+1)^2&s=1$ then $latex f^2(x) = ((x+1)^2+1)^2&s=1$ and so on. Don’t get this notation mixed up with exponentiation where for example if $latex g(x) = x + 1&s=1$ then $latex g^2(x) = x+2&s=1$ and $latex [g(x)]^2=(x+1)^2&s=1$. When working with certain functions there are exceptions to this notation for repeated composition. For example $latex \sin^2 x&s=1$ means $latex (\sin x)^2&s=1$ and not $latex \sin(\sin x) &s=1$ as it probably should. Mathematicians do not always practice what they preach. I am also guilty of using this flawed notation, but perhaps there will come a time when this is all straightened out.

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