Functions in Mathematics

The concept of functions in mathematics is presented with definitions, examples, practice problems, and detailed solutions.

Definition of a Function in Mathematics

A function from set \( D \) to set \( R \) is a relation that assigns to each element \( x \) in \( D \) exactly one element \( y \) of \( R \). The set \( D \) is the domain (inputs) and set \( R \) is the range (outputs).

Functions Represented by Venn Diagrams

Figures 1 and 2 show Venn diagrams representing functions because each input in \( D \) corresponds to exactly one output in \( R \).

Function representation using Venn diagram
Fig.1 - Function represented by Venn diagram
Another function representation using Venn diagram
Fig.2 - Another function represented by Venn diagram

Note: Two different inputs (like 3 and 5) having the same output (0) does not violate the function definition.

Relation that is not a function
Fig.3 - Not a function (input 6 has two outputs)

Functions Represented by Ordered Pairs

Question

Which relations represent functions?

  1. \( R_1 = \{ (a,2), (b,4), (c,7), (d,-2) \} \)
  2. \( R_2 = \{ (a,4), (a,5), (b,5), (c,5) \} \)
  3. \( R_3 = \{ (0,4), (2,4), (4,5), (6,5) \} \)

Solution

  1. All domain elements are different ⇒ \( R_1 \) is a function.
  2. Input \( a \) has two different outputs ⇒ \( R_2 \) is not a function.
  3. All domain elements are different ⇒ \( R_3 \) is a function.

Functions Represented by Graphs

Question

Explain why these relations are/aren't functions:

  1. Fig.4: Not a function - input \( x = 2 \) has outputs 4 and 6.
    Graph that is not a function
    Fig.4 - Not a function
  2. Fig.5: Is a function - each input has exactly one output.
    Graph that is a function
    Fig.5 - Is a function
  3. Fig.6: Not a function - multiple inputs have multiple outputs.
    Another graph that is not a function
    Fig.6 - Not a function

Functions Represented by Equations

Examples: \( y = 2x - 1 \), \( y = x^2 + 1 \), \( y = \dfrac{1}{x} \). All represent \( y \) as function of \( x \) because each \( x \) value produces exactly one \( y \) value. Here \( x \) is the independent variable and \( y \) the dependent variable.

More on functions represented by equations.

Domain and Range of a Function

The domain \( D \) is the set of all input values for which outputs are defined.

The range \( R \) is the set of all output values corresponding to inputs in \( D \).

Example

Find domain and range for these representations:

a) Ordered Pairs

\( F_1 = \{ (a,2), (b,4), (c,7), (d,-2) \} \)
Domain \( D = \{ a, b, c, d \} \)
Range \( R = \{ 2, 4, 7, -2 \} \)

b) Venn Diagram

Venn diagram function for domain/range example
Fig.7 - Function as Venn diagram

Domain \( D = \{ 0, 5, 3, -3 \} \)
Range \( R = \{ 2, 0, 6 \} \)

c) Graph

Graph function for domain/range example
Fig.8 - Function as graph

Domain \( D = \{ 2, 4, 6, 7 \} \)
Range \( R = \{ 3, 1, 4, 2 \} \)

d) Equation

\( y = -x + 3 \) with \( D = \{-3, 0, 6, 7\} \)
Calculations:
\( x = -3 ⇒ y = 6 \)
\( x = 0 ⇒ y = 3 \)
\( x = 6 ⇒ y = -3 \)
\( x = 7 ⇒ y = -4 \)
Range \( R = \{ 6, 3, -3, -4 \} \)

Practice Problems

(Solutions)

Part A: Venn Diagrams

Venn diagrams for practice problems
Fig.9 - Relations as Venn diagrams

Which relations are functions? Explain.

Part B: Graphs

Graphs for practice problems
Fig.10 - Relations as graphs

Which graphs represent functions? Explain.

Part C: Ordered Pairs

Which relations are functions?

  1. \( R_1 = \{ (-2,2), (0,4), (9,7), (12,-2) \} \)
  2. \( R_2 = \{ (a,0), (a,7), (a,9), (a,5) \} \)
  3. \( R_3 = \{ (0,4), (2,4), (5,4), (9,4) \} \)
  4. \( R_4 = \{ (a,b), (b,a), (c,e), (d,a) \} \) (assuming all letters have different values)

Part D: Multiple Representations

For \( y = x - 2 \) with domain \( D = \{-1, 0, 1, 2\} \), represent as:

  1. Table
  2. Ordered pairs
  3. Graph
  4. Venn diagram

Part E: Domain and Range

  1. From Fig.11:
    Venn diagram for domain/range problem
    Fig.11 - Function as Venn diagram
  2. \( y = \dfrac{1}{x+2} \) with \( D = \{0, 1, 4\} \)
  3. From Fig.12:
    Graph for domain/range problem
    Fig.12 - Function as graph

Solutions

Part A Solutions

  1. Each input has exactly one output ⇒ is a function.
  2. Input 3 has outputs A and B ⇒ is NOT a function.
  3. Each input has one output ⇒ is a function.
  4. Input 5 has outputs a, b, and c ⇒ is NOT a function.

Part B Solutions

  1. Each \( x \) has one \( y \) ⇒ is a function.
  2. \( x = 1 \) has \( y = 2 \) and \( y = -2 \) ⇒ is NOT a function.
  3. \( x = 1 \) has four \( y \) values ⇒ is NOT a function.
  4. Each \( x \) has one \( y \) ⇒ is a function.

Part C Solutions

  1. All inputs are different ⇒ is a function.
  2. Input \( a \) has multiple outputs ⇒ is NOT a function.
  3. All inputs are different ⇒ is a function.
  4. All inputs are different ⇒ is a function.

Part D Solutions

  1. Table:
    \( x \)-1012
    \( y \)-3-2-10
  2. Ordered pairs: \( \{ (-1,-3), (0,-2), (1,-1), (2,0) \} \)
  3. Graph:
    Graph of y = x - 2
    Fig.13 - Graph of \( y = x - 2 \)
  4. Venn diagram:
    Venn diagram of y = x - 2
    Fig.14 - Venn diagram of \( y = x - 2 \)

Part E Solutions

  1. Domain \( D = \{1,3,8,12,23\} \), Range \( R = \{-4,4\} \)
  2. Domain \( D = \{0,1,4\} \), Range \( R = \{\frac{1}{2}, \frac{1}{3}, \frac{1}{6}\} \)
  3. Domain \( D = \{-1,1,3,4,6\} \), Range \( R = \{-2,1\} \)

References

Algebra and Trigonometry - Swokowsky Cole - 1997 - ISBN: 0-534-95308-5
Algebra and Trigonometry with Analytic Geometry - R.E.Larson, R.P. Hostetler, B.H. Edwards, D.E. Heyd - 1997 - ISBN: 0-669-41723-8
Relations in Mathematics
Functions Represented by Equations
Free Online Tutorials on Functions and Algebra