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Exercises Functions

Example 2 from Algebra and Trigonometry, 3.4 Composition of Functions

Determining whether Composition of Functions is Commutative

Using the functions provided, find $f (g (x))$ and $g (f (x)) .$ Determine whether the composition of the functions is commutative.

f (x) = 2 x + 1 g (x) = 3 - x

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
Let’s begin by substituting $g (x)$ into $f (x) .$

$\begin{matrix} f (g (x)) & = & 2 (3 - x) + 1 \\ = & 6 - 2 x + 1 \\ = & 7 - 2 x \end{matrix}$
Now we can substitute $f (x)$ into $g (x) .$

$\begin{matrix} g (f (x)) & = & 3 - (2 x + 1) \\ = & 3 - 2 x - 1 \\ = & −2 x + 2 \end{matrix}$

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

How did it go?

PreviousPerforming Algebraic Operations on Functions Next Interpreting Composite Functions

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 3.4, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.