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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.