What is a Function?
A function is a rule that assigns exactly one output to each input from a set called the domain. Functions are written using notation like f(x), and they are the backbone of algebra, calculus, and modeling real-world relationships such as distance over time or cost per item.
A function f is a relation between a set of inputs (the domain) and a set of outputs (the range) where every input maps to exactly one output, commonly written as y = f(x).
Try it: interactive calculator
Step-by-step worked examples
If f(x) = 2x + 1, find f(3).
Substitute x = 3 into the function f(3) = 2(3) + 1 f(3) = 7
Is the relation {(1,2), (1,5), (2,3)} a function?
Check if any input repeats with different outputs Input 1 maps to both 2 and 5 Since one input has two outputs, this is NOT a function
If g(x) = x² − 4, find the input(s) where g(x) = 0.
Set the function equal to 0: x² − 4 = 0 Solve: x² = 4 x = 2 or x = −2
Flashcards
Quick quiz
Q1.If f(x) = 3x − 2, what is f(4)?
Q2.Which relation is NOT a function?
Q3.What test determines if a graph represents a function?
Q4.For f(x) = x², what is the range?
The full card deck, worked steps and AI-tutor support for “What is a Function?” are in Notek — study by hand before your exam.
Common mistakes
Thinking f(x) means f multiplied by x. — Correct: f(x) is function notation meaning 'the output of f for input x', not multiplication.
Believing every equation is a function. — Correct: An equation is a function only if each input has exactly one output (e.g. x = y² is not a function).
Confusing domain and range. — Correct: Domain = valid inputs (x-values); range = resulting outputs (y-values).
Assuming a repeated output means it's not a function. — Correct: Repeated outputs are fine — two different inputs CAN share the same output; only repeated inputs with different outputs break the rule.
FAQ
What is a function in math?
A function is a rule where every input value produces exactly one output value, written as f(x).
What is the formula for a linear function?
f(x) = mx + b, where m is the slope and b is the y-intercept.
What are examples of functions?
f(x) = 2x + 1, g(x) = x², and h(x) = √x are all common examples of functions.
How do you calculate the output of a function?
Substitute the given input value for x in the function's rule and simplify.