Everyday Usage of "Composition" and usage in mathematics

In everyday language, the word composition refers to the combination of different elements to form a whole. For example, in art, a composition is the arrangement of visual elements in a work. In writing, it refers to the way ideas, sentences, and paragraphs are combined to create a cohesive text.

In mathematics, composition refers specifically to the combination of two functions to form a new function. This process, known as function composition, involves applying one function to the result of another function. Function composition is represented by the notation \( (f \circ g)(x) \), which means applying the function \( g \) to \( x \) and then applying the function \( f \) to the result of \( g(x) \). The expression $f \circ g(x)$ is read "$f$ of $g$ of $x$." The function $f \circ g$ is sometimes called a composite function.

The term composition is used to indicate that two functions are being put together to create a third function.

Importance in Mathematics

If we have two functions \( f(x) \) and \( g(x) \), the composite function \( f(g(x)) \) is created by plugging the output of \( g(x) \) into \( f(x) \). This new function \( f(g(x)) \) gives us a way to combine the effects of two different functions into a single operation.

Notice that the composition $f \circ g(x)$ only makes sense if $g(x)$ is something that can sensibly plugged into $f$. So when you write an expression of the form $f \circ g(x)$, you should make sure that the domain of $f$ contains the range of $h$.

Function composition is particularly useful in cases where one quantity depends on another, and that second quantity depends on a third one. For instance, consider a scenario where you want to calculate the cost of heating a house based on the day of the year. Here, the temperature on a specific day determines the heating cost, and the day of the year determines the temperature. By composing these two functions, you can directly find the heating cost based on the day of the year.

Exercises