MATH1202 Chap.3 Relations, Functions and Transformations
Relations, Functions and Transformations
This chapter builds the language needed to read every later graph. It distinguishes relations from functions, finds natural domains and ranges, evaluates and composes functions, tests even and odd symmetry, and moves landmark points under transformations. The central habit is to carry the rule and its permitted inputs together rather than treating domain as an afterthought.
Algebraic rules, mapping descriptions and graphs are treated as coordinated views of the same object. Inputs rejected by a denominator, root or logarithm are recorded before further operations. Transformation work begins with parent landmarks, which makes the effect of inside and outside changes testable rather than dependent on a memorised picture.
What this chapter covers
- 01
Vertical-line function test
- 02
Natural domain restrictions
- 03
Range from graph and rule
- 04
Function notation and composition
- 05
Even and odd symmetry
- 06
Parent graphs and landmark transformations
Worked example · free
Transform a parent parabola
- +1Identify shifts, reflection and vertical stretch.
- +1Read vertex and axis from vertex form.
- +1Calculate horizontal and vertical intercepts.
- +1State the range and check symmetry.
Key terms
- Relation
- A set of ordered input-output pairs.
- Function
- A relation assigning exactly one output to each permitted input.
- Composition
- Applying one function to the output of another function.
- Transformation
- A systematic translation, reflection or scale change applied to a graph.
Relations, Functions and Transformations FAQ
Can two inputs have the same output?
Yes. A function requires each input to have one output, but different inputs may share that output. The vertical-line test checks repeated outputs for one input; the horizontal-line test is instead related to one-to-one behaviour and inverses.
Why do horizontal shifts use the opposite sign?
The input expression reaches the parent landmark when the inside equals the parent input. For x minus h to equal zero, x must equal h, so the parent origin moves right to h.
How should I combine domain restrictions?
Write each condition separately, such as a non-negative radicand and a non-zero denominator, then take their intersection. The final domain must satisfy every condition at the same time.
What must be checked in a composition?
For a composition such as f of g of x, the input must first belong to the domain of g, and the output from g must belong to the domain of f. Simplifying the formula alone can conceal one of these restrictions. Write the input-output chain, state both conditions, and test any excluded boundary in the unsimplified composition.
How are even and odd symmetry tested algebraically?
Replace x by negative x without changing any other sign. If the simplified result equals the original rule, the function is even and has y-axis symmetry. If it equals the negative of the original rule, it is odd and has origin symmetry. A function may be neither, and its domain must also be symmetric for either label.
Why are landmark points safer than tracing a graph by eye?
A landmark carries exact coordinates through a transformation. Inside changes alter the input coordinate, while outside changes alter the output coordinate. Moving the intercepts, endpoints, turning points or asymptotic reference points gives an auditable skeleton. A freehand sketch can then join those features while respecting domain, range and the known overall shape.
How do I tell whether a relation is one-to-one?
First confirm it is a function using the vertical-line test or the rule. Then ask whether distinct permitted inputs can share an output. A horizontal line meeting the graph more than once shows it is not one-to-one. Algebraically, set equal outputs and determine whether equality forces the two inputs to be equal.
What information should accompany a transformed graph?
Name the parent family, list the horizontal and vertical changes, and label enough exact landmarks to determine placement and scale. State domain, range, intercepts and asymptotes or endpoints when relevant. A curve drawn without these features is difficult to verify. Substitute at least one moved landmark into the final rule and ensure the displayed restrictions agree with the algebraic domain.
Exam move
Create a small parent-graph library with labelled landmarks, domains, ranges and any asymptotes. For every transformed graph, list inside and outside changes, move at least two landmarks, and state domain and range. Use different colours for input changes and output changes until the order becomes automatic.
For compositions, draw an input-output chain, apply the inner rule first and track which values are permitted at each stage. For domains, write restrictions separately and intersect them on a number line. Mix symbolic, tabular and graphical questions so the definition of function is not tied to one presentation.
Verify a transformed landmark by substitution into the final rule, then compare the completed sketch with the transformed domain and range.
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