Functions
The Functions domain tests the idea that a function pairs each allowed input with exactly one output. You will interpret function notation, connect equations, tables, and graphs, identify domain and range, and recognize linear, quadratic, and exponential behavior.
Core rules
- A relation is a function if no input has two different outputs; the vertical line test detects failures on a graph.
- f(a) means substitute a for every x. Evaluate from the inside out for compositions such as f(g(2)).
- Domain restrictions: exclude inputs that make a denominator zero or an even root negative.
- Linear functions have constant slope m = (y₂ − y₁)/(x₂ − x₁); quadratic functions form parabolas; exponential functions change by a constant factor.
- Transformations: f(x) + k shifts up; f(x − h) shifts right; a·f(x) stretches or reflects; f(−x) reflects across the y-axis.
- Average rate of change over [a, b] is [f(b) − f(a)]/(b − a).
Worked example
For f(x) = 2x² − 3x + 1, find f(−2) and the domain of g(x) = 1/(x − 4). Substitute −2: 2(−2)² − 3(−2) + 1 = 2(4) + 6 + 1 = 8 + 6 + 1 = 15. For g, the denominator cannot be zero, so x − 4 ≠ 0; the domain is all real numbers except x = 4. On a number line that is (−∞, 4) ∪ (4, ∞).
Traps that cost marks
- Treating f(x + 1) as f(x) + 1; the input changes, not the output.
- Reversing horizontal shifts: f(x − 3) moves the graph right, not left.
- Ignoring domain restrictions hidden in denominators, radicals, or context.
- Confusing slope with a y-intercept, or computing slope with points in inconsistent order.
- Assuming exponential growth is slower than quadratic growth; eventually exponentials overtake polynomials.