S. D. V. Stephens

notes / Integrated Mathematics / chapter 8: Equations, Functions and Graphs

8.2 Functions: Definition, Notation and Domain

What a function is, how to tell whether a relation is one, and how to evaluate a function, find its domain and combine it with others.

Functions

Here is the dependent variable and is the independent variable.

Definition 8.2.1. A function is a relationship that matches 1 input to 1 output.

  • Input: usually the -values.
  • Output: the or values.

An input never gives more than 1 output.

For example, matching work hours to pay:

Mapping diagram from work hours to pay: 25 to $500, 53 to $310, 30 to $490 and 40 to $490; a dashed arrow from 30 to $310 is crossed out, since one input may not have two outputs.

The dashed arrow from to is crossed out: already goes to , and an input can’t have a second output.

Note.

  • Domain: , the set of inputs (’s).
  • Range: , the set of outputs (’s).

D: input. R: output.

is a function: does not repeat.

is a non-function: does repeat ( appears twice; that’s the bad one).

Solving each equation for :

  • : function.
  • : function.
  • : non-function (the is the problem).
  • : function.
  • : non-function.
  • : function.
  • : non-function.

The vertical line test

For ’s, 1 input gives 1 output. A vertical line picks out one input, so it can’t hit the graph of a function more than once.

Vertical line test: (a) input and output axes with a vertical line at one input; (b) a line and (c) a parabola meet every vertical line once, so they are functions; (d) a sideways parabola, (e) a circle and (f) a vertical line meet some vertical line more than once, so they are non-functions.
  • (a) Output axis vertical, input axis horizontal; a vertical line stands over one input.
  • (b) A line: sweep vertical lines left and right, and each hits it once: function.
  • (c) A parabola: function.
  • (d) A sideways parabola: vertical lines hit it twice: non-function.
  • (e) A circle: non-function.
  • (f) A vertical line: non-function.

Evaluating functions

In function notation, becomes , and becomes .

To evaluate for :

which gives the point .

Example 8.2.2.

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Example 8.2.3.

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under our real domain (it’s a complex #).

Example 8.2.4.

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Difference quotient

Definition 8.2.5. The difference quotient is

Sketch for the difference quotient: a line through the points at heights f(x) and f(x+h) on the vertical axis, whose horizontal separation is h.

Example 8.2.6. Find the difference quotient of .

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Finding the domain

Definition 8.2.7. The domain is the set of inputs of a function that give a real number output.

Three problem areas:

  1. Square roots : the inside must be nonnegative (positive or ).
  2. Denominators: cannot equal .
  3. Logs, which you will learn later.

Example 8.2.8.

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We need :

Domain: , or .

Example 8.2.9.

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We need . Find where it is :

so , , or . Thus

or .

Example 8.2.10.

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, or .

Example 8.2.11.

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, or .

Example 8.2.12.

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We need and .

or .

Example 8.2.13.

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We need and , so .

Example 8.2.14.

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We need and , so

or .

Operations on functions

Sums, differences, products and quotients of two functions, with the domain of each.

Example 8.2.15. ,

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Both have , since .

since only when , i.e. .

Example 8.2.16. ,

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has and has .