DPST1012 Chap.2 Functions, Domain, Range and Graphs
Functions, Domain, Range and Graphs
The definition that decides everything
Week 2 opens the first topic, Functions, with a single definition: a function maps every element of an input set to exactly one element of an output set. The worksheets test it on everyday tables such as menus, accommodation prices, step counts and class timetables.
Two inputs may share an output, as when two drinks cost the same, but one input may never have two outputs, which is why a timetable with two classes in one slot fails. Inputs and outputs can be numerical or non numerical.
Notation, domain and range
Function notation f : x ↦ f(x) and the equation y = f(x) say the same thing, and questions move between them.
The domain is the set of inputs and the range is the set of outputs actually produced. Sets are unordered, so the order in which a range is listed does not matter.
For a formula with no context, the domain is every real number the formula accepts, which means excluding zeros of denominators and negative values under square roots; in context the domain is usually smaller.
Graphs, interpolation and the line tests
The graph of a function is its set of ordered pairs plotted on a number plane.
Joining neighbouring points is interpolation and turns a table into a continuous model that can be read in two directions: up from an input to find an output, or across from an output to find every input that produces it. The vertical line test checks whether a curve is a function at all, and the horizontal line test checks whether it is one-to-one, the property that matters again when inverse functions arrive in Week 4.
What this chapter covers
- 01
A function as a mapping between two sets
- 02
Numerical and non numerical inputs
- 03
Function notation and y = f(x)
- 04
Domain and range from tables and formulas
- 05
Graphs as sets of ordered pairs
- 06
Interpolation and extrapolation
- 07
One-to-one functions and the line tests
Worked example · free
Decide the domain and range of two rules and evaluate them
- 1The root in p needs x + 1 ≥ 0, so the domain of p is x ≥ −1.
- 1√(x + 1) takes every value from 0 upwards, so p(x) = 5 − √(x + 1) takes every value up to 5: range p(x) ≤ 5.
- 2q is undefined at x = 3, so its domain is every real number except 3, and since 2/(x − 3) is never 0 its range is every real number except 0.
- 1p(8) = 5 − √9 = 2 and q(1) = 2/(1 − 3) = −1.
Key terms
- Function
- A rule that assigns to every element of an input set exactly one element of an output set.
- Domain
- The set of all input values for which a function is defined or which the context allows.
- Range
- The set of output values that a function actually produces from its domain.
- Ordered Pair
- A pair of numbers written (x, y) in which the order matters, used to plot one input with its output.
- One-to-One Function
- A function in which every output comes from exactly one input, so it passes the horizontal line test.
- Interpolation
- Estimating a value between two known data points, usually by joining them with a line or curve.
Functions, Domain, Range and Graphs FAQ
Can two different inputs give the same output?
Yes, and the rule is still a function. A menu where two dishes cost the same price is a valid function. What is not allowed is a single input with two different outputs.
How do I find the domain of a formula quickly?
Look for a denominator and remove the x-values that make it zero, then look for an even root and require its contents to be zero or positive. Anything else a polynomial or exponential accepts is usually fine.
What is the difference between the vertical and horizontal line tests?
A vertical line crossing a graph more than once means some input has two outputs, so the graph is not a function. A horizontal line crossing more than once means an output repeats, so the function is not one-to-one.
Why do worksheets ask me to change times to 24-hour format?
Times such as 1:00 could mean morning or afternoon, and a function needs unambiguous inputs. A 24-hour clock gives every time of day a single numerical label that can be plotted on an axis.
Why does a graph need a break in its axis?
When every output is far from zero, such as temperatures between 19 and 27 degrees, starting the axis at zero squashes the data into a thin band. A marked break lets the scale start near the smallest value while warning the reader that zero has been skipped.
Exam move
For every table you meet, say aloud which column is the input and why, write the domain and range as sets, and then answer one 'evaluate' and one 'solve' question from it. Keep a list of the questions you misread, because reading direction is most of this topic.
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