Functions: notes and practice questions

A function maps each input to exactly one output (one-to-one or many-to-one). Vertical Line Test: If a vertical line intersects a graph more than once, it is not a function.

Sketch: Produce a rough graph outline, clearly label all key points (intercepts, minimums/maximums, asymptotes). Draw: Produce an accurate graph on a scaled grid, plot specific points from GDC, join with a straight line or smooth curve.

x-intercepts (roots/zeros): Points where $y=0$. y-intercept: Point where $x=0$.

Modelling: Simplifies real-world situations for prediction/analysis. Domain restrictions: Always consider real-life context (e.g., $t \ge 0$, $x > 0$, $0 \le t < 24$).

Mathematical Model: Simplifies real-world situations for mathematical description and prediction. Assumptions: Simplify mathematics (e.g., ignoring air resistance).

Gradient $m$ measures slope: $m = \frac{y_2 - y_1}{x_2 - x_1}$ for points $(x_1, y_1)$ and $(x_2, y_2)$. Gradient-intercept form: $y = mx + c$, where $m$ is gradient and $(0, c)$ is y-intercept.

A composite function $(f \circ g)(x)$ or $f(g(x))$ applies one function to the output of another; the function closest to $x$ is applied first. For $(f \circ g)(x)$ to be valid, the range of $g(x)$ must be within the domain of $f(x)$.

Transformations alter position, orientation, or size; use "stretch" with a scale factor. Horizontal translation by vector $\begin{pmatrix} a \\ 0 \end{pmatrix}$: $y = f(x - a)$, $(x, y) \to (x + a, y)$, VA: $x=k \to x=k+a$.

Mathematical model: simplifies real-world situations for mathematical description and prediction. Initial Value: starting value when independent variable (e.g., $t$) is zero.

Logarithmic scales convert exponentially increasing values (e.g., 1, 10, 100) to a manageable linear scale (e.g., 0, 1, 2) by using logarithms. Semi-log graphs have one logarithmic axis and one linear axis.