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Exercises Functions

Example 1 from Algebra and Trigonometry, 3.4 Composition of Functions

Performing Algebraic Operations on Functions

Find and simplify the functions $(g - f) (x)$ and $(\frac{g}{f}) (x),$ given $f (x) = x - 1$ and $g (x) = x^{2} - 1.$ Are they the same function?

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
Begin by writing the general form, and then substitute the given functions.

$\begin{matrix} (g - f) (x) & = & g (x) - f (x) \\ (g - f) (x) & = & x^{2} - 1 - (x - 1) \\ (g - f) (x) & = & x^{2} - x \\ (g - f) (x) & = & x (x - 1) \\ (\frac{g}{f}) (x) & = & \frac{g (x)}{f (x)} \\ (\frac{g}{f}) (x) & = & \frac{x^{2} - 1}{x - 1} \\ (\frac{g}{f}) (x) & = & \frac{(x + 1) (x - 1)}{x - 1} & where x \neq 1 \\ (\frac{g}{f}) (x) & = & x + 1 \end{matrix}$
No, the functions are not the same.

Note: For $(\frac{g}{f}) (x),$ the condition $x \neq 1$ is necessary because when $x = 1,$ the denominator is equal to 0, which makes the function undefined.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Find and simplify the functions $(f g) (x)$ and $(f - g) (x) .$

f (x) = x - 1 and g (x) = x^{2} - 1

Are they the same function?

How did it go?

PreviousFinding Absolute Maxima and Minima from a Graph Next Determining whether Composition of Functions is Commutative

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.