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Lesson 6: Operations on functions

Topic. This lesson covers Session 6: Operations on functions

Learning Outcomes.

  • Compute and simplify the sum, difference, product and quotient of functions.
  • Compute and simplify the composition of functions.

WeBWorK. There is 1 WeBWorK assignment on today’s material:

  1. Functions – Operations

Additional Video Resources.

Question of the Day: If we add two functions together, is the result a function?

Operations on functions

We can make new functions by combining functions in various ways — these include the usual operations (addition, subtraction, multiplication, division) and an important new idea called composition.

Example 6.1. Let $f(x)=x^{2}+5 x$ and $g(x)=7 x-3$. Find the following functions, and state their domain.
$$
(f+g)(x),(f-g)(x),(f \cdot g)(x), \text { and }\left(\frac{f}{g}\right)(x)
$$

Example 6.3. Let $f(x)=\sqrt{x+2},$ and let $g(x)=x^{2}-5 x+4 .$ Find the functions $\frac{f}{g}$ and $\frac{g}{f}$ and state their domains.

VIDEO: Examples – Operations on Functions

Video by Irania Vazquez

Composition of functions

The composition of functions describes what happens when we put an input into an initial function $g$, and then we plug the output of $g$ into another function $f$ — it is like plugging a value into two functions in a row

Definition. Let $f$ and $g$ be functions, and assume that $g(x)$ is in the domain of $f .$ Then define the composition of $f$ and $g$ at $x$ to be
$$
(f \circ g)(x):=f(g(x))
$$

Example. Let $f(x)=x^{2}+1$ and $g(x)=x+3 .$ Find the following compositions
a) $(f \circ g)(3)$
b) $(g \circ f)(3)$
c) $(f \circ g)(x)$
d) $(g \circ f)(x)$

VIDEO: Example – Composition of functions

Video by Irania Vazquez

Exit Question

Given the functions $f(x)=x^2+3x$ and $g(x)=\sqrt{x}$, find and simplify the compositions $(f\circ g)(x)$ and $(g\circ f)(x)$.

Answer

$(f\circ g)(x) = x+3\sqrt{x}$ and $(g\circ f)(x) = \sqrt{x^2+3x}$.

Good job! You are now ready to practice on your own – give the WeBWorK assignment a try. If you get stuck, try using the “Ask for Help” button to ask a question on the Q&A site.