Introduction to Functions
Understand function notation, how to evaluate functions, and the concepts of domain and range.
What is a Function?
A function is a rule that assigns exactly one output to each input. Think of it as a machine: you put a number in, and get exactly one number out.
Input x = 3, f(x) = 2x + 1, so f(3) = 2(3) + 1 = 7.
Key point: For something to be a function, every input must produce exactly one output. If an input gives two different outputs, it is NOT a function.
Function Notation
Instead of writing y = 2x + 1, we can write f(x) = 2x + 1. Here, f(x) is read as "f of x". The letter f names the function, and x is the input variable.
Evaluating a Function
To find f(4) when f(x) = 3x − 2:
Replace x with 4: f(4) = 3(4) − 2 = 12 − 2 = 10
f(x) = x² + 1, find f(3)
f(3) = 3² + 1 = 9 + 1 = 10
g(x) = 5 − 2x, find g(−1)
g(−1) = 5 − 2(−1) = 5 + 2 = 7
Domain and Range
Domain
The domain is the set of all allowed input values (x-values). For most functions, the domain is all real numbers unless there is a restriction (e.g., cannot divide by zero or take the square root of a negative number).
Range
The range is the set of all possible output values (y-values). The range depends on the function rule and the domain.
For f(x) = 2x + 1 with domain {1, 2, 3}: f(1)=3, f(2)=5, f(3)=7. Range = {3, 5, 7}.
Key Vocabulary
| Term | Definition |
|---|---|
| Function | A rule that assigns exactly one output value to each input value. |
| Function notation | Writing f(x) to represent a function named f with input variable x. |
| Domain | The set of all allowed input values (x-values) for a function. |
| Range | The set of all possible output values (y-values) produced by a function. |
Worked Examples
Evaluating a Function
If f(x) = 4x − 3, find f(5) and f(−2).
f(5): Replace x with 5. f(5) = 4(5) − 3 = 20 − 3 = 17
f(−2): Replace x with −2. f(−2) = 4(−2) − 3 = −8 − 3 = −11
Finding the Range from a Domain
For g(x) = x² − 1 with domain {−2, 0, 1, 3}, find the range.
g(−2) = (−2)² − 1 = 4 − 1 = 3
g(0) = 0² − 1 = −1
g(1) = 1² − 1 = 0
g(3) = 3² − 1 = 8
Range = {−1, 0, 3, 8}
Is it a Function?
A mapping shows: 1 → 4, 2 → 7, 3 → 4, 2 → 9. Is this a function?
Check: Input 2 maps to both 7 AND 9.
Conclusion: This is NOT a function because input 2 has two different outputs.
In a function, each input must have exactly one output.
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see feedback.
Question 1
If f(x) = 3x + 2, what is f(4)?
Question 2
If g(x) = x² − 4, what is g(−3)?
Question 3
What is the domain of a function?
Question 4
For h(x) = 2x − 1 with domain {1, 2, 3}, what is the range?
Question 5
A mapping has: 1 → 5, 2 → 8, 3 → 11. Is this a function?
Key Concepts Summary
- ●A function gives exactly one output for each input.
- ●Function notation: f(x) means "f of x". Evaluate by substituting the input value for x.
- ●The domain is the set of all valid input values; the range is the set of all output values.
- ●Two inputs producing the same output is fine; one input producing two outputs is NOT a function.