Relations and Functions
What you must understand
A relation is any set of ordered pairs; a function is a relation in which each input (x-value) has exactly one output (y-value). Use the vertical-line test on a graph and check for repeated x-values with different y-values in a table or set. Know function notation f(x), and be able to evaluate, interpret, and solve equations such as f(x) = k.
Understand domain and range, and how they connect to real contexts (time cannot be negative, a population is not fractional). Recognize families of functions: linear, quadratic, exponential, and their graphs. Linear functions have a constant rate of change (slope); exponential functions have a constant ratio. Be able to identify whether data is best modelled as linear or exponential by checking differences versus ratios.
Traps to watch for
- Calling a relation a function when an x-value repeats with different y-values.
- Confusing slope (rate of change) with the y-intercept (initial value).
- Mixing up domain (inputs) and range (outputs).
- Forgetting to restrict domain in context problems.
- Assuming a graph is exponential without checking that successive values share a constant ratio.
Rule of thumb
Linear: constant differences. Exponential: constant ratios. Check the data before choosing a model.
Worked example: For f(x) = 2x + 3, f(4) = 2(4) + 3 = 11. To solve f(x) = 15: 2x + 3 = 15, so x = 6.