Functions — Objectives | Mathematics Department

Function Notation and Representation

1. Understand the definition of a function as a rule that assigns each input exactly one output.
2. Evaluate a function for a given input (e.g. find $f(3)$ given $f(x) = 2x - 5$).
3. Solve equations of the form $f(x) = k$ to find inputs that give a chosen output.
4. Substitute algebraic expressions into a function (e.g. find $f(a+1)$ or $f(2x)$).
5. Move between numerical, algebraic, table and graphical representations of a function.

6. Identify the domain and range of a function from its graph.
7. Find a sensible (natural) domain when a function is defined by a formula, including avoiding division by zero or square roots of negatives.
8. Sketch and interpret linear functions, including those written as $f(x) = mx + c$.
9. Sketch and interpret piecewise linear functions defined on different intervals.
10. Use the vertical line test to decide whether a graph represents a function.

11. Form a composite function $f \circ g$ and evaluate it for given inputs.
12. Recognise that composition is not commutative ($f \circ g \neq g \circ f$ in general).
13. Find the inverse $f^{-1}(x)$ of a linear function algebraically.
14. Recognise that the graph of $y = f^{-1}(x)$ is the reflection of $y = f(x)$ in the line $y = x$.
15. Verify an inverse by showing $f(f^{-1}(x)) = x$.