Functions and Graphs
Core concepts
Know domain, range, injectivity, monotonicity, parity (even/odd), periodicity, and inverse functions. Be able to sketch and transform graphs: shifts f(x ± a), scalings, reflections, and |f(x)|. Understand composition, the relationship between a function and its inverse (reflection across y = x), and standard graphs including linear, quadratic, cubic, rational, absolute value, exponential, logarithmic, and trigonometric functions.
Traps
- Reporting the domain of a composite function without intersecting with the inner function's domain.
- Assuming f⁻¹ exists whenever f is defined; only one-to-one functions on their domain have inverses.
- Confusing horizontal shifts with vertical ones, or reversing the direction of f(x − a).
- Forgetting that the range of f becomes the domain of f⁻¹.
- Missing that |f(x)| reflects negative portions above the axis, creating new turning points.
Worked example
Find the inverse of f(x) = (2x − 1)/(x + 3). Write y(x + 3) = 2x − 1, so yx + 3y = 2x − 1, giving x(y − 2) = −1 − 3y, hence x = (−1 − 3y)/(y − 2). Therefore f⁻¹(x) = (−1 − 3x)/(x − 2), with domain x ≠ 2 (the range of f).
Rule of thumb
To find an inverse, swap x and y and solve for y; then state the domain explicitly as the original function's range.