3.4 Composition of functions  (Page 3/9)

Let’s begin by substituting $g\left(x\right)$ into $f\left(x\right).$

Now we can substitute $f\left(x\right)$ into $g\left(x\right).$

We find that $g(f(x))\ne f(g(x)),$ so the operation of function composition is not commutative.

The inside expression in the composition is $s(3).$ Because the input to the s -function is time, $t=3$ represents 3 minutes, and $s(3)$ is the number of sit-ups completed in 3 minutes.

Using $s(3)$ as the input to the function $c(s)$ gives us the number of calories burned during the number of sit-ups that can be completed in 3 minutes, or simply the number of calories burned in 3 minutes (by doing sit-ups).

The function $y=f\left(x\right)$ is a function whose output is the number of miles driven corresponding to the number of hours driven.

The function $g\left(y\right)$ is a function whose output is the number of gallons used corresponding to the number of miles driven. This means:

The expression $g(y)$ takes miles as the input and a number of gallons as the output. The function $f(x)$ requires a number of hours as the input. Trying to input a number of gallons does not make sense. The expression $f\left(g(y)\right)$ is meaningless.

The expression $f(x)$ takes hours as input and a number of miles driven as the output. The function $g(y)$ requires a number of miles as the input. Using $f(x)$ (miles driven) as an input value for $g(y),$ where gallons of gas depends on miles driven, does make sense. The expression $g\left(f(x)\right)$ makes sense, and will yield the number of gallons of gas used, $g,$ driving a certain number of miles, $f(x),$ in $x$ hours.