A function is a rule that assigns exactly one output to each input. Function notation, written as f(x), names the function f and shows it's being applied to the input x — so f(x) = 2x + 3 means "take x, double it, then add 3." Evaluating a function means substituting a specific value in for x: f(4) means substitute 4 for x, giving f(4) = 2(4) + 3 = 11.
Example
A phone plan costing $20 plus $0.50 per gigabyte of data used could be modelled as f(g) = 20 + 0.5g, where g is gigabytes used. To find the cost for 10GB, calculate f(10) = 20 + 0.5(10) = $25.
Key terms
Function:
A rule assigning exactly one output to each input.
f(x) notation:
Notation showing a function f is applied to input x.
Domain:
The set of all possible input values for a function.
Questions
1. A function assigns:
Exactly one output to each input
Multiple random outputs to one input
No outputs at all
Only negative outputs
2. f(x) means:
The function f applied to input x
Multiplying f by x
Subtracting x from f
A type of punctuation
3. If f(x) = 2x + 3, then f(4) equals:
11
8
9
14
4. If f(x) = x + 5, then f(0) equals:
5
0
10
-5
5. The domain of a function is:
The set of all possible input values
Only the outputs
A type of graph paper
The function's name only
6. If f(x) = 3x, then f(5) equals:
15
8
3
35
7. Evaluating a function means:
Substituting a specific value in for x
Deleting the function entirely
Ignoring the input value
Renaming the function only
8. If f(x) = 4x - 1, then f(3) equals:
11
12
9
13
9. If g(x) = x² + 2, then g(3) equals:
11
9
6
8
10. For the phone plan f(g) = 20 + 0.5g, the cost for 10GB is:
$25
$20
$30
$25.50
11. If f(x) = 2x - 5, and f(x) = 9, then x equals:
7
4
2
9
12. If h(x) = x/2 + 1, then h(6) equals:
4
3
7
6
13. Why is function notation like f(x) considered more useful than just writing y, especially with multiple functions?
It clearly labels which function and input are being used, avoiding confusion between multiple equations
f(x) notation makes equations impossible to solve
y and f(x) can never represent the same kind of relationship
Function notation removes the need for any input value
14. If f(x) = x² - 4, then f(-2) equals:
0
-8
8
4
15. Why must a function assign only one output to each input, unlike some general relations?
This is the defining property of a function — without it, an input's outcome would be ambiguous
Functions are allowed to have unlimited outputs for one input
One-output-per-input is only a suggestion, not a requirement
This property applies only to linear functions, not others
16. If f(x) = 2x + 1 and g(x) = x - 3, what is f(g(5))?
5
3
11
9
17. Why might modelling a real-world cost (like a phone plan) as a function be useful for decision-making?
It allows quick, systematic calculation of outcomes for any input value, like data usage
Functions can never be used to model real-world costs
Modelling costs as functions always produces inaccurate results
This approach only works for costs that never change
18. If f(x) = 3x + 2 for all x in the domain {1, 2, 3}, what is the range of f?
{5, 8, 11}
{3, 6, 9}
{1, 2, 3}
{2, 5, 8}
19. Why is restricting a function's domain sometimes necessary in real-world modelling (e.g. g cannot be negative in the phone plan example)?
Some inputs would be meaningless or impossible in the real context being modelled
Restricting domain always makes a function invalid
Every real-world function must accept every possible number
Domain restrictions have no connection to real-world contexts
20. If f(x) = x² and g(x) = f(x) + 3, what is g(4)?
19
16
13
7
21. Why might function notation be useful when comparing two different pricing plans, like f(x) for Plan A and g(x) for Plan B?
It allows both plans to be evaluated and compared clearly at the same input value
Function notation makes it impossible to compare two different plans
Only one function can ever be defined at a time
Comparing plans requires ignoring their mathematical relationship entirely
Answer key (parent copy)
1. Exactly one output to each input
2. The function f applied to input x
3. 11
4. 5
5. The set of all possible input values
6. 15
7. Substituting a specific value in for x
8. 11
9. 11
10. $25
11. 7
12. 4
13. It clearly labels which function and input are being used, avoiding confusion between multiple equations
14. 0
15. This is the defining property of a function — without it, an input's outcome would be ambiguous
16. 5
17. It allows quick, systematic calculation of outcomes for any input value, like data usage
18. {5, 8, 11}
19. Some inputs would be meaningless or impossible in the real context being modelled
20. 19
21. It allows both plans to be evaluated and compared clearly at the same input value