MATH1011 Chap.5 Functions, Domains and Composition
Functions, Domains and Composition
A function includes its allowed inputs
A function is not merely a formula. It consists of an input set, an output set and a rule that assigns exactly one output to each allowed input. Two inputs may share an output without causing trouble; one input may not produce two different outputs.
Changing the domain changes the function even when every visible symbol in the rule stays the same.
The natural domain is the largest real input set on which the formula makes sense. For the functions used here, the main restrictions are immediate: an even root needs a nonnegative expression underneath it, a denominator cannot be zero, and a logarithm needs a positive argument. Apply every restriction together.
If a forbidden denominator value lies inside the interval allowed by a root, the domain has a puncture and must be written in two pieces.
Range is produced, codomain is declared
The codomain is the set nominated to receive outputs, while the range is the set of outputs the function actually reaches. A codomain can be larger than the range.
To find a range, use the graph, solve y = f(x) for possible inputs, or transform a known standard range. Endpoints deserve separate attention: a closed domain end may create an attained range end, while a value approached but never reached stays open.
Standard graphs act as reusable shapes.
Absolute value forms a V, a square forms an upward parabola, the reciprocal has two branches with forbidden axes, and the square root begins at its endpoint. A horizontal shift changes the input before the rule acts, while a vertical shift changes the output after it acts.
Read those roles rather than memorising an isolated picture.
Composition is a pipeline with a domain check
The composition g after f means apply f first and feed that result into g. Order matters because reversing the machines generally changes the answer and may change whether the expression is defined. The domain of the composition contains only inputs accepted by f whose f-values are accepted by g.
Check both stages before simplifying the combined formula, especially when roots, denominators or logarithms appear.
Piecewise functions demand the same discipline repeatedly. Evaluate the first branch, then use the resulting value to decide the branch on the next pass. The branch is chosen from the current input at that stage, not from the original number.
Write each intermediate output on its own line so that a sign change or boundary value cannot silently send the calculation down the wrong rule.
What this chapter covers
- 01
Function rules, domains and codomains
- 02
Natural-domain restrictions from roots and denominators
- 03
Range as the set of outputs actually reached
- 04
Standard graphs and geometric transformations
- 05
Composition order and the pipeline test
- 06
Repeated evaluation of piecewise functions
Worked example · free
Build a composition and keep only the legal inputs
- 2Apply f first and impose its square-root restriction.
- 2Feed the result into g and identify the forbidden output of f.
- 2Combine the restrictions and write the domain in interval notation.
Key terms
- Domain
- The Domain is the set of inputs a function is allowed to receive.
- Codomain
- The Codomain is the declared set in which every output is required to lie.
- Range
- The Range is the subset of the codomain that the function actually reaches.
- Composition
- Composition sends the output of one function into another, with the inner function acting first.
- Piecewise Function
- A Piecewise Function uses different rules on stated parts of its domain while still assigning one value to each input.
Functions, Domains and Composition FAQ
Why is a formula not enough to identify a function?
The same formula can define different functions when its allowed inputs or declared output set changes. A rule such as the square root on all nonnegative real numbers is not the same function as the identical rule restricted to one interval, even where their numerical outputs agree.
How do I find the domain of a composition?
First find which inputs the inner function accepts. Then require every resulting inner value to lie in the outer function's domain. Combine those two conditions before simplifying, because an algebraically neat combined formula can hide a value rejected at an intermediate stage.
What is the practical difference between range and codomain?
The codomain is chosen in advance and may contain values never produced. The range is discovered from the rule and domain, so it records only outputs that actually occur. A range question therefore requires calculation or graph reading rather than copying the set written after the arrow.
Exam move
For every function problem, write D, rule and range as separate lines before manipulating the expression. On compositions, draw two boxes and put each restriction under the box that creates it. Rehearse the standard absolute-value, square, reciprocal and square-root graphs from blank axes, then describe each shift in words before drawing it.
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