Functions

Learn the definition of a function as a mapping $f: A \to B$ that assigns each element of the domain exactly one element of the codomain, and how to identify functions from tables and graphs using the vertical line test.

Learn the $f(x)$ notation for writing and reading functions, interpret expressions like $f(3)$ and $f(a)$ as specific outputs, and use different letters such as $f$, $g$, and $h$ to distinguish between multiple functions.

Learn how to evaluate a function at a specific number or algebraic expression by substituting into the function rule, and how to find the input that produces a given output.

Learn how to identify the domain of a function by finding values that cause division by zero or square roots of negatives, and express the domain using interval notation.

Learn what the range of a function is and how it differs from the codomain.

Learn how to compose two functions using the notation $f \circ g$, evaluate compositions at specific inputs, and determine when a composition is defined by checking that the output of the inner function lies in the domain of the outer function.

Understand what an inverse function is, how the domain and range swap, and how to verify an inverse using composition.

Learn how to find inverse functions algebraically by swapping and solving, apply the procedure to linear functions, and understand why the graph of an inverse is a reflection across the line $y = x$.

Learn what it means for a function to be one-to-one, identify one-to-one functions using the definition or counterexamples, and apply the horizontal line test to determine whether a function is one-to-one from its graph.

Learn why a function must be one-to-one to have an inverse, use the horizontal line test to determine invertibility, and make non-invertible functions invertible by restricting their domains.

Learn how to classify a function as even, odd, or neither using the algebraic test $f(-x) = \pm f(x)$, and read the same parity off a graph by recognising $y$-axis or origin symmetry.

Learn how to read and evaluate piecewise function definitions by applying the rule whose condition matches the input, and sketch piecewise graphs using closed and open dots to mark whether each endpoint is included.