2.3 Functions, Domain, Range, Graph

1. Definition of a Function

A function $f$ from a set $X$ to a set $Y$ assigns to each element $x$ of $X$ a unique element $y$ of $Y$. The notation is $f: X \rightarrow Y$ or $f(x) = y$.

EXAMPLE 1 & 2 (Identifying Functions)

If $X = \{1, 2, 3\}$ and $Y = \{a, b, c, d\}$:
Mapping $1 \mapsto a$, $2 \mapsto b$, $3 \mapsto d$ is a valid function.
Mapping $1 \mapsto a$, $2 \mapsto b$, $3 \mapsto b$ is a valid function (multiple elements can map to the same image).
However, mapping $1 \mapsto a$, $2 \mapsto c$ (where 3 is left unmapped) is not a function.
Mapping $1 \mapsto a$, $2 \mapsto b$ AND $2 \mapsto c$ is not a function (one input cannot have multiple outputs).

2. Domain and Range

Domain ($D_f$): The set of all valid $x$ inputs.
Range ($R_f$): The set of all resulting $y$ outputs.

A function mapping $x$ to its double is defined as $f: \mathbb{R} \rightarrow \mathbb{R}$, given by $f(x) = 2x$. If constrained to the interval $[0, 10]$, the domain is $x \in [0, 10]$ and the resultant range becomes $y \in [0, 20]$.

The Domain "Agreement"

If the domain is unspecified, it is assumed to be the largest possible subset of $\mathbb{R}$. Key restrictions include:

  1. Polynomials have no restrictions: $D_f = \mathbb{R}$.
  2. Fractions $f(x) = A / B$ restrict the denominator: $B \ne 0$.
  3. Square roots $f(x) = \sqrt{A}$ restrict the radicand: $A \ge 0$.
  4. Logarithms $f(x) = \log A$ restrict the argument: $A > 0$.

EXAMPLE 5

(a) $f(x) = 3x - 9 \Rightarrow D_f: x \in \mathbb{R}$.
(b) $f(x) = \frac{5}{3x - 9} \Rightarrow 3x - 9 \ne 0 \Rightarrow D_f: x \in \mathbb{R} \setminus \{3\}$.
(c) $f(x) = \sqrt{3x - 9} \Rightarrow 3x - 9 \ge 0 \Rightarrow D_f: x \in [3, +\infty)$.
(d) $f(x) = \ln(3x - 9) \Rightarrow 3x - 9 > 0 \Rightarrow D_f: x \in (3, +\infty)$.
(e) $f(x) = \frac{x+2}{x^2 - 3x + 2} \Rightarrow x^2 - 3x + 2 \ne 0 \Rightarrow x \ne 1, 2$.
(f) $f(x) = \sqrt{x-1} + \sqrt{2-x} \Rightarrow x \ge 1$ and $x \le 2 \Rightarrow D_f: x \in [1, 2]$.
(g) $f(x) = \frac{\sqrt{1-x^2}}{x} \Rightarrow 1-x^2 \ge 0$ and $x \ne 0 \Rightarrow D_f: x \in [-1, 0) \cup (0, 1]$.

3. Graphical Characteristics

  • Vertical Line Test: Determines if a graph represents a function. A vertical line must intersect the graph at most once.
  • y-intercept: Evaluate $f(0)$.
  • x-intercepts (Roots): Solve $f(x) = 0$.
  • Intersection Points: Solve $f(x) = g(x)$ for two graphs.

EXAMPLE 6 & 7

Given $f(x) = (x-3)^2 - 4$ and $g(x) = x - 5$:

Roots of f: $(x-3)^2 - 4 = 0 \Rightarrow x = 5$ or $x = 1$.
Min of f: Occurs at the vertex $V(3, -4)$.
Intersections: $f(x) = g(x) \Rightarrow (x-3)^2 - 4 = x - 5 \Rightarrow x^2 - 7x + 10 = 0 \Rightarrow x = 2$ and $x = 5$. Intersection coordinates are $(2, -3)$ and $(5, 0)$.
Inequality $f(x) > g(x)$: The graph of $f$ is above $g$ outside the roots of $f(x) - g(x) = 0$. Therefore, $x < 2$ or $x > 5$.

EXAMPLE 8

Solve $2^x = 2x + 3$.

Solution: Solving graphically or via GDC tools (SolveN) yields the roots $x \approx -1.30$ and $x \approx 3.25$. To solve $2^x < 2x + 3$, look for where the exponential curve sits below the line, yielding the interval $-1.30 < x < 3.25$.

4. One-to-One and Onto Functions (HL)

One-to-One (Injective): Different inputs yield different outputs: $x_1 \ne x_2 \Rightarrow f(x_1) \ne f(x_2)$. Evaluated using the Horizontal Line Test (horizontal lines cross at most once).
Onto (Surjective): The range completely coincides with the co-domain set $Y$. Any element of $Y$ is mapped.

EXAMPLE 9 & 10

$f(x) = 2x$ is one-to-one (passes horizontal line test) and onto (range is full $\mathbb{R}$).
$f(x) = x^2$ over $\mathbb{R}$ is many-to-one (fails horizontal line test since $f(2) = f(-2)$) and not onto (range is strictly $[0, +\infty)$).