Functions and graphs
What this domain tests
This domain examines whether you can describe a function completely: domain, range, symmetry, monotonicity, asymptotes, inverse and transformations, and link each feature to its graph. Questions often combine an algebraic expression with a sketch or a parameter.
Core rules
- Domain restrictions: denominators cannot be zero, radicands of even roots must be non-negative, and logarithm arguments must be positive.
- Composition: (f o g)(x) = f(g(x)); the domain of the composition excludes values where g(x) is undefined or outside the domain of f.
- Inverse: write y = f(x), swap x and y, solve for y. The domain of f^(-1) equals the range of f, and their graphs are reflections in y = x. A function has an inverse only if it is one-to-one on its domain.
- Transformations: f(x) + k shifts up by k; f(x + h) shifts left by h; af(x) stretches vertically by a; f(bx) compresses horizontally by 1/b. Apply horizontal changes in the order written.
- Asymptotes: for a rational function, cancel common factors first; remaining zeros of the denominator give vertical asymptotes, and end behaviour gives horizontal or oblique asymptotes.
- Parity: f is even if f(-x) = f(x) for all x, odd if f(-x) = -f(x).
Worked example
For f(x) = (2x - 1)/(x + 3), find the inverse and the asymptotes. The domain is x not equal to -3. Set y = (2x - 1)/(x + 3), so y(x + 3) = 2x - 1. Rearranging: yx + 3y = 2x - 1, then x(y - 2) = -1 - 3y, giving x = (3y + 1)/(2 - y). Thus f^(-1)(x) = (3x + 1)/(2 - x), defined for x not equal to 2, which matches the range of f (y not equal to 2). The graph of f has vertical asymptote x = -3 and horizontal asymptote y = 2; the inverse has x = 2 and y = -3, as expected from reflection.
Traps that cost marks
- Writing f^(-1) as 1/f is wrong; the inverse is a different function.
- Forgetting to restrict the inverse domain, or to check that f is one-to-one before inverting.
- Applying a horizontal stretch before a horizontal shift when the notation requires the reverse.
- Reading the range from the visible window rather than from the algebraic behaviour.