Relations and Functions: Models, Transformations, and Inverses
Core concepts
Relations and Functions tests your ability to represent relationships symbolically, graphically, and numerically. You must recognise function families - linear, quadratic, exponential, logarithmic, polynomial, rational, radical, and trigonometric - and match a context or data pattern to the correct model. Know domain and range restrictions, function notation, composition, and inverses. A function has an inverse function only if it is one-to-one; otherwise you must restrict the domain.
Transformations follow (y = af(b(x-h))+k): (a) stretches vertically and reflects over the x-axis if negative, (b) compresses horizontally and reflects over the y-axis, (h) shifts horizontally (opposite sign), and (k) shifts vertically. Regression work requires interpreting correlation, residuals, and the coefficient of determination.
Traps to watch
- Applying horizontal shifts in the wrong direction.
- Forgetting that (f^{-1}(x)) is not (1/f(x)); swap x and y and solve.
- Ignoring domain restrictions introduced by square roots or denominators.
- Confusing exponential growth (constant percent change) with linear growth (constant amount change).
- Over-extrapolating a regression beyond the data range.
Rule of thumb
Test a model by checking whether the rate of change is constant (linear), the ratio is constant (exponential), or the second difference is constant (quadratic).
Worked example
Find the inverse of (f(x)=2x-6). Write (y=2x-6), swap: (x=2y-6), solve: (y=\frac{x+6}{2}). So (f^{-1}(x)=\frac{x+6}{2}), with domain all reals.