4.1 Function Notation, Domain, and Range

Why this matters on Math 30-2

Relations and Functions is the largest Math 30-2 content area in the current Alberta bulletin, so the exam can ask about graphs, formulas, tables, and applied models in several formats. Before you can decide whether a model is polynomial, rational, exponential, logarithmic, or sinusoidal, you must know what the inputs mean, what outputs are possible, and whether the relation passes the function test.

A relation is any set of ordered pairs. A function is a relation where each input has exactly one output. The same output may appear more than once, but the same input cannot lead to two different outputs. On a graph, this is the vertical line test: if any vertical line hits the graph more than once, the graph is not a function.

Function notation

Function notation is compact, but it is often where diploma errors begin.

Do not read f(2) as multiplication. It is not f times 2. It means the value of the function when the input is 2. If f(x) = -2x + 7, then f(3) = -2(3) + 7 = 1. If the question says f(x) = 1, then you solve -2x + 7 = 1, so x = 3.

Domain and range

The domain is the set of allowable inputs. The range is the set of possible outputs. For a table, list the input and output values shown. For a graph, read left to right for domain and bottom to top for range. For a formula, start with algebraic restrictions, then apply context.

Common algebraic restrictions include denominators that cannot equal zero and square roots that cannot have negative radicands. In Math 30-2 rational work, the most common restriction is a non-permissible value from a denominator. For example, g(x) = (x + 1)/(x - 4) has domain x != 4 because x = 4 makes the denominator zero. That restriction exists before you think about graph shape, intercepts, or simplification.

Context can add limits. A height model for a thrown object may be defined only while the object is in the air. A revenue model may use whole-number tickets only. A graphing calculator may draw a smooth curve, but the real situation may allow only selected inputs.

Worked example: interpreting a context model

A ball is launched from the ground and its height is modeled by h(t) = -2(t - 5)^2 + 50, where t is time in seconds and 0 <= t <= 10.

1. The input is time, so the domain is 0 <= t <= 10.
2. The vertex form shows the maximum height is 50 at t = 5.
3. At t = 0 and t = 10, h(t) = 0, so the least height in the modeled interval is 0.
4. The range is 0 <= h(t) <= 50.

The algebraic parabola continues forever, but the diploma question gave a time interval. Using all real numbers for the domain would ignore the physical context and would produce impossible negative heights after the flight is over.

Reading from graphs and tables

When a graph is shown, mark the extreme x-values and y-values that are actually included. Use filled endpoints as included values and open circles as excluded values. When a table is shown, do not assume missing inputs are allowed unless the question states a pattern or model. A table of measured values may be discrete; a formula for a continuous path may fill the values between measurements.

Diploma traps

• Confusing f(4) with the x-value where f(x) = 4.
• Giving the domain of a simplified rational expression but forgetting restrictions from the original expression.
• Treating every graph as continuous when the context is tickets, people, games, or objects that must be counted in whole numbers.
• Reporting the y-intercept as a domain value. The y-intercept is an output point when x = 0.
• Ignoring a stated interval such as 0 <= t <= 10 because the calculator displays more of the curve.

Quick procedure

1. Identify what the input represents.
2. Identify what the output represents.
3. Check whether each input has exactly one output.
4. Find algebraic restrictions from the original rule.
5. Apply practical restrictions from the context.
6. State domain and range in words, inequalities, interval notation, or listed values as the question requires.

Numerical-response habit

Numerical-response items do not give four choices to catch a notation mistake. If the question asks for f(6), write the substitution line before entering numbers: f(6) = expression with 6 replacing x. If the question asks for the input when f(x) = 6, write the equation first. This one written line separates an evaluation task from a solving task.

For range questions, check whether the output is a maximum, minimum, or listed set. A restricted quadratic may have a range such as 2 <= y <= 18, while a table may have a range such as {4, 7, 9}. Use the notation that best matches the representation.