Properties of Functions and Their Graphs
Properties of Functions and Their Graphs
This domain tests domain and range, function notation, composition, inverses, symmetry, transformations, intercepts, asymptotes, and piecewise functions. Questions often require reasoning from an equation or stated properties rather than from a visible graph.
Core rules
- Domain restrictions: exclude values that make a denominator zero; require even-index radicands to be nonnegative; require logarithm arguments to be positive.
- Vertical line test: a relation is a function if no vertical line meets its graph more than once.
- Composition: (f ∘ g)(x) = f(g(x)). The domain of the composition excludes x-values not in g’s domain and values where g(x) is not in f’s domain.
- Inverses: swap x and y, then solve for y. The domain of f becomes the range of f⁻¹, and vice versa. A function has an inverse function only if it passes the horizontal line test.
- Transformations: f(x − h) + k shifts right h and up k. The sign inside the parentheses behaves opposite to intuition.
- Symmetry: even functions satisfy f(−x) = f(x) and are symmetric about the y-axis; odd functions satisfy f(−x) = −f(x) and are symmetric about the origin.
- Asymptotes: vertical asymptotes come from excluded domain values; horizontal asymptotes describe end behavior and depend on degree comparison.
Worked example
Find the inverse of f(x) = (2x − 1)/(x + 3).
Write y = (2x − 1)/(x + 3), then swap x and y: x = (2y − 1)/(y + 3). Multiply: x(y + 3) = 2y − 1, so xy + 3x = 2y − 1. Collect y-terms: xy − 2y = −1 − 3x, so y(x − 2) = −1 − 3x. Thus y = (−1 − 3x)/(x − 2) = (1 + 3x)/(2 − x). The inverse is f⁻¹(x) = (1 + 3x)/(2 − x), with x ≠ 2. This matches the domain-range swap: f has a horizontal asymptote y = 2, which becomes a vertical asymptote x = 2 for f⁻¹.
Traps that cost marks
- Confusing f⁻¹(x) with 1/f(x).
- Reversing the horizontal shift: f(x − 3) shifts right, not left.
- Forgetting to restrict the inverse’s domain to the range of the original function.
- Ignoring values excluded by the inner function in a composition.
- Treating a removable discontinuity as a vertical asymptote without checking cancellation.
- Applying the even/odd test with arithmetic errors, such as miscomputing (−x)² or (−x)³.