Definition of a Function
Domain and Range
Function Notation
Types of Functions
Linear Functions
Linear functions are the simplest and most fundamental type. They also form the basis for solving systems of equations.
Quadratic Functions
See our detailed quadratic equations guide for more.
Rational Functions
See our detailed rational functions guide for more.
Exponential Functions
See our exponents and logarithms guide for more.
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Inverse Functions
Function Composition
Common Mistakes
Thinking f(x) means f times x
f(x) is notation for "the function f evaluated at input x." It is not multiplication. This confusion causes errors throughout algebra.
Confusing horizontal and vertical transformations
Horizontal transformations act opposite to what you might expect: f(x-2) shifts right, not left. Vertical transformations are more intuitive.
Practice Problems
Related Guides
Frequently Asked Questions
What is a function in math?
A function is a rule that assigns exactly one output to each input. For every x-value in the domain, there is exactly one y-value. The vertical line test is a quick graphical check: if any vertical line crosses the graph more than once, it is not a function.
What is the difference between domain and range?
The domain is the set of all valid input values (x-values) for a function. The range is the set of all possible output values (y-values) the function produces. For example, f(x) = √x has domain [0, ∞) and range [0, ∞).
What does f(x) mean?
f(x) is function notation. It names the function (f) and shows the input variable (x). f(3) means "evaluate the function f at input 3." It does not mean f times x — this is one of the most common beginner misunderstandings.
What is a composite function?
A composite function applies one function to the result of another. Written as (f ∘ g)(x) = f(g(x)), it means "first apply g to x, then apply f to the result." The order matters: f(g(x)) is usually different from g(f(x)).
How do you find the inverse of a function?
To find the inverse, swap x and y in the equation and solve for y. The inverse function "undoes" what the original function does. Not all functions have inverses — a function must be one-to-one (pass the horizontal line test) to have an inverse.
What are transformations of functions?
Transformations shift, stretch, compress, or reflect a function's graph. Common transformations include vertical/horizontal shifts (adding constants), vertical/horizontal stretches (multiplying), and reflections (negating). Understanding transformations lets you graph any function family from its parent function.
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