Functions
The Functions domain asks you to interpret functions in graphs, tables, equations, and verbal descriptions. MCAS Grade 10 emphasizes function notation, domain and range, average rate of change, transformations, and choosing between linear, quadratic, and exponential models.
Core rules
- Function notation: f(a) means substitute a for every x in the rule. f(g(x)) is composition, evaluated inside first.
- Domain restrictions: exclude values that make a denominator zero; for even roots, require the radicand to be nonnegative.
- Average rate of change on [a, b] is (f(b) − f(a))/(b − a), the slope of the secant line.
- Transformations: f(x) + k shifts vertically; f(x − h) shifts horizontally in the opposite direction of h; a·f(x) stretches or compresses vertically; f(−x) reflects across the y-axis.
- Linear functions have a constant rate of change; exponential functions have a constant ratio; quadratic functions have constant second differences.
- Increasing, decreasing, and positive/negative intervals are read from left to right on the graph.
Worked example
Let f(x) = x² − 2x and g(x) = 3x + 1. Find f(g(2)). Work inside out: g(2) = 3(2) + 1 = 7. Then f(7) = 7² − 2(7) = 49 − 14 = 35. Composition is not multiplication; f(g(2)) is not f(2)·g(2).
Traps
- f(x + h) is not f(x) + h; substitute the entire input.
- Horizontal shifts go opposite the sign inside the parentheses.
- Do not confuse f(g(x)) with g(f(x)); order matters.
- A graph can cross the x-axis and still be a function; the vertical-line test is the check.