Functions: notes and practice questions

A straight line can be expressed in different forms: Gradient-Intercept Form:

A function maps each input to exactly one output. Domain: The set of all possible input values ($x$) for which the function is defined.

Involves understanding the key features of functions to produce accurate representations. Key concepts:

Key properties: max/min points, x-/y-axis intercepts, symmetry, vertex (quadratics), and asymptotes (rational/exponential functions). Found using derivatives, algebra, or GDC tools.

A composite function combines two functions $f(x)$ and $g(x)$: $(f \circ g)(x) = f(g(x))$.

General form: $f(x) = ax^2 + bx + c$. Vertex form: $f(x) = a(x-h)^2 + k$, where $(h, k)$ is the vertex.

Factorising: Write $ax^2 + bx + c = 0$ as $(px + q)(rx + s) = 0$. Completing the square (CTS): Rewrite $ax^2 + bx + c$ as $a(x-h)^2 + k$.

Reciprocal functions: $f(x) = \frac{1}{x}$ or similar. Rational functions: $f(x) = \frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomials.

Exponential functions: $f(x) = a \cdot b^x$ ($b > 0, b \neq 1$). Key features: growth/decay, horizontal asymptote at $y = 0$.

Solve $f(x) = 0$: Find roots/zeros where the graph intersects the x-axis. Solve $f(x) = g(x)$: Find intersections of the two functions.

Translation: Shifts the graph vertically ($f(x) + k$) or horizontally ($f(x - h)$). Reflections: Across the x-axis ($-f(x)$) or y-axis ($f(-x)$).

Polynomials are expressions of the form $a_n x^n + \dots + a_0$. The degree is $n$, and $a_n$ is the leading coefficient. The Division Algorithm states $P(x) = Q(x)D(x) + R(x)$, where the degree of $R(x)$ is less than the degree of $D(x)$.

Rational functions are ratios of polynomials, $f(x) = P(x)/Q(x)$. Domain excludes roots of $Q(x)$. Intercepts are found by setting $Q(x)=0$ (x-intercepts) and $f(0)$ (y-intercept).

Odd Functions: Symmetric about the origin ($f(-x) = -f(x)$). Examples: $x^3$, $\sin x$. Even Functions: Symmetric about the y-axis ($f(-x) = f(x)$). Examples: $x^2$, $\cos x$.

Inequalities can be solved graphically using a GDC by plotting functions and finding intersections, or analytically. Quadratic inequalities can be solved by sketching the parabola based on its roots and concavity, or by using a sign diagram.

The absolute value function $|x|$ is defined as $x$ for $x \geq 0$ and $-x$ for $x < 0$, resulting in a V-shaped graph with its vertex at the origin. Transformations of the form $y = a|x - h| + k$ involve vertical stretching/reflection ($a$), and translations of the vertex to $(h, k)$.