5.01 Functions and relations
What is the name used to describe a graph where for some value of x, there exists two or more different values of y?
Determine whether the following statements are true or false:
When working with a function, substituting a certain value of x into the formula gives only 1 value of y for that value of x.
A horizontal line can intersect the graph of a function at more than one point.
A relation always passes the vertical line test.
All functions are relations.
State whether each of the following statements are correct:
Some relations are functions.
All functions are relations.
No functions are relations.
No relations are functions,
State whether each of the following statements are correct:
The graph of every non-vertical straight line is a function.
There is no straight line graph that is a function.
The graph of every straight line is a relation.
The graph of every non-horizontal straight line is a function.
Determine whether each of the following are:
a function,
a relation,
a relation and a function, or
neither
\left\{\left(2, 5\right), \left(7, - 3 \right), \left(5, 2\right), \left( - 4 , - 9 \right)\right\}
\left\{\left(2, 5\right), \left(2, 7\right), \left( - 3 , - 4 \right), \left( - 9 , 13\right)\right\}
(1,5), (1,1), (7,-2), (-5,-10)
(1,5), (7,-2), (-5,-10), (13,-13)
(1,5), (1,7), (-2,-5), (-5,-10)
y = 9 x
y = x^{2} + 2
y = \dfrac{3}{x}
y^{2} + x^{2} = 25
y^{2} = x
y = - x^{3}
State whether the values in the tables below represent a function:
| x | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|---|---|---|
| y | -4 | -3 | -2 | -1 | 0 | -1 | -2 | -3 | -4 |
| x | 0 | 1 | 4 | 8 | 9 | 12 | 16 | 18 | 20 |
|---|---|---|---|---|---|---|---|---|---|
| y | 0 | 1 | 2 | \dfrac{2}{\sqrt{2}} | 3 | \dfrac{2}{\sqrt{3}} | 4 | \dfrac{3}{\sqrt{2}} | \dfrac{2}{\sqrt{5}} |
| x | -9 | -7 | -6 | -5 | -3 | -2 | 3 | 5 | 10 |
|---|---|---|---|---|---|---|---|---|---|
| y | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 |
| x | -3 | -2 | -2 | 0 | 1 | 1 | 2 |
|---|---|---|---|---|---|---|---|
| y | 8 | 4 | 3 | 7 | 2 | 1 | 0 |
Consider the points in the table.
| x | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|---|---|---|
| y | 4 | 3 | 2 | 1 | 0 | 1 | 2 | 3 | 4 |
Plot the points on a number plane.
Do they represent a function?
Write down the coordinates of all the points that would need to be removed so that the remaining points form an increasing function.
Consider the points in the table:
| x | -4 | -3 | 0 | -1 | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|---|---|---|
| y | 4 | 3 | 2 | 1 | 0 | 1 | 2 | 3 | 4 |
Plot the points on a number plane.
Do they represent a function?
A relation is defined as follows:
y = 1 if x is positive and y = -1 if x is 0 or negative.
Complete the table for this relation.
| x | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|---|---|---|
| y |
Plot the points on a number plane.
Do these values represent a function?
Consider these ordered pairs:
\left\{\left( - 9 , - 5 \right), \left( - 5 , - 10 \right), \left( - 5 , - 4 \right), \left( - 3 , 7\right), \left( - 2 , - 4 \right), \left( - 1 , 1\right)\right\}
Plot the ordered pairs on a number plane.
Which ordered pair would need to be removed from the set so that the remaining ordered pairs represent a function?
Write down a value of k such that the relation \left\{\left(6, 2\right), \left(8, 5\right), \left(1, 7\right), \left(k, 4\right)\right\} does not represent a function.
Does there exist any values of k such that the relation \left\{\left(2, 1\right), \left(7, k\right), \left(5, 8\right), \left(3, 9\right)\right\} does not represent a function?
CheapCalls Mobile charges \$1.10 a minute plus a connection fee of 70c for an international call.
Complete the table:
Is this relation a function?
| Call length (minutes) | International Call cost (dollars) |
|---|---|
| 1 | |
| 2 | |
| 3 | |
| 4 | |
| 5 |
A particular store is offering one free T-shirt for every two T-shirts purchased, where each T-shirt costs \$19.
Complete the table:
Is this relation a function?
| Number of T-shirts | Total Cost (dollars) |
|---|---|
| 1 | |
| 2 | |
| 3 | |
| 4 | |
| 5 |
A particular internet service provider charges \$40 a month on their 20 GB plan plus \$10 for each additional 10 GB used.
Complete the table:
Is this relation a function?
| Total GB Used | Total Charge (dollars) |
|---|---|
| 10 | |
| 20 | |
| 30 | |
| 40 | |
| 50 |