# Relations and Functions

# Relations

A relation is any set of points that connect two variables.

A relation is often expressed as an equation connecting the variables $x$ and $y$. The relation is a set of points $(x,y)$ in the Cartesian plane. This plane is separated into four quadrants according to the signs of $x$ and $y$.

For example, $y=x+1$ and $x=y^2$ are the equations of two relations. Each equation generates a set of ordered pairs, which we can graph:

$\cdots,(-2,-1),(-1,0),(0,1),(1,2),(2,3),\cdots$

$\cdots,(4,2),(1,1),(0,0),(1,-1),(4,-2),\cdots$

# Functions

A function, sometimes called a mapping, is a relation in which two different ordered pairs now have the same $x$-coordinate or first component.

We can see from the above definition that a function is a special type of relation.

Every function is a relation, but not every relation is a function.

## Testing for Functions

**Algebraic Test:** If a relation is given as an equation, and the substitution of any value for $x$ results in one and only one value of $y$, then the relation is a function.

For example,

$y=x+1$ is a function, as for any value of $x$ there is only one corresponding value of $y$.

- $x=-2 \rightarrow y=-1$
- $x=-1 \rightarrow y=0$
- $x=0 \rightarrow y=1$
- $x=1 \rightarrow y=2$
- $x=2 \rightarrow y=3$

$x=y^2$ is not a function since if $x=4$ then $y=\pm 2$. That is because one $x$ value corresponds to two $y$ values.

- $x=4 \rightarrow y=2$
- $x=4 \rightarrow y=-2$
- $x=1 \rightarrow y=1$
- $x=1 \rightarrow y=-1$

**Vertical Line Test or Geometric Test:** If we draw all possible vertical lines on the graph of a relation, the relation:

- is
**a function**if each line cuts the graph no more than once

- is
**not a function**if at least one line cuts the graph more than once

### Example 1

Is the set of ordered pairs ${(1,4),(2,5),(3,6),(4,7)}$ a function?

${(1,4),(2,5),(3,6),(4,7)}\rightarrow$ a function as for any value of $x$ there is only one corresponding value of $y$.

### Example 2

Is the set of ordered pairs ${(1,3),(2,2),(3,4),(4,2)}$ a function?

${(2,2),(4,2)}\rightarrow$ not a function as one $x$ value is corresponding two $y$ values.

### Example 3

Is the set of ordered pairs ${(0,0),(1,1),(2,2),(3,3)}$ a function?

${(0,0),(1,1),(2,2),(3,3)}\rightarrow$ a function as for any value of $x$ there is only one corresponding value of $y$.

### Example 4

Is the set of ordered pairs ${(0,0),(1,0),(2,0),(3,0)}$ a function?

${(0,0),(1,0),(2,0),(3,0)}\rightarrow$ a function as for any value of $x$ there is only one corresponding value of $y$.

### Example 5

Is the set of ordered pairs ${(1,0),(1,1),(1,2),(1,3)}$ a function?

${(1,0),(1,1),(1,2),(1,3)}\rightarrow$ not a function as one $x$ value is corresponding two $y$ values.

### Example 6

Is the graph a function?

The graph is a function as each vertical line cuts the graph only once.

### Example 7

Is the graph a function?

The graph is not a function, as each vertical line cuts the graph more than once.

Algebra Algebraic Fractions Arc Binomial Expansion Capacity Chain Rule Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Measurement Perfect Square Perimeter Prime Factorisation Probability Product Rule Proof Pythagoras Theorem Quadratic Quadratic Factorise Ratio Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume

## Responses