P3 1.2 - Functions domain and range

P3 1.2 - Functions domain and range

Describe a function using its rule, its allowed inputs and its possible outputs. Use domains, graphs and algebra to distinguish one-one from many-one mappings and to find complete ranges.

One-one and many-one mappings

Can squaring be a function when 22 and −2-2 give the same answer? Yes: the essential condition concerns the output from each input, rather than the number of inputs sharing an output.

The rule “square the input” maps −2,−1,0,1,2-2,-1,0,1,2 to 4,1,0,1,44,1,0,1,4. Each input has exactly one output. Writing f(x)=x2f(x)=x^2 names this rule; f(2)=4f(2)=4 says that input 22 produces output 44.

Function and mapping types

A function assigns exactly one output to each input in its domain, the allowed set of inputs. Its range is the set of outputs actually attained.

A one-one function gives different outputs for different inputs. A many-one function allows distinct inputs to have the same output. Both are functions.

Squaring on all real inputs is many-one: f(−2)=f(2)f(-2)=f(2). Restricting the inputs to x≥0x\ge0 makes it one-one, because nonnegative inputs with equal squares must be equal.

[DIAGRAM: asset_name: Squaring with two domains; asset_slug: squaring-mappings; caption: Removing negative inputs makes the squaring mapping one-one.; description: Two arrow diagrams. On the left inputs negative 2, negative 1, 0, 1 and 2 map to outputs 4, 1 and 0 under squaring, with pairs sharing 4 and 1. On the right inputs 0, 1 and 2 map to 0, 1 and 4 with no shared outputs.]
Removing negative inputs makes the squaring mapping one-one.

A rule “send 44 to either 22 or −2-2” is not a function unless a single output is selected. The square-root function selects the nonnegative root, so 4=2\sqrt4=2, rather than ±2\pm2.

For a graph, a vertical line must meet it at most once for it to represent yy as a function of xx. A horizontal line meeting it twice shows that the function is many-one. Thus a full parabola passes the vertical-line test but fails the one-one test.

Key idea

One output for each allowed input makes a function. Repeated outputs make it many-one, not invalid.

Distinguish repeated outputs from multiple outputs for one input.

Domain restrictions

A formula can restrict its own inputs, and a stated domain can restrict them further.

Over the real numbers, division by zero is undefined and a square root requires a nonnegative quantity inside it. If no domain is stated, find the largest real set on which the rule makes sense. If a domain is stated, use that set: do not silently enlarge it.

Domain restrictions

For a real function, a denominator must be nonzero and the expression under a square root must be at least zero. When the square root is itself in a denominator, its value must be positive.

For example, 1/(x2−9)1/(x^2-9) requires x≠−3,3x\ne-3,3, while 9−x2\sqrt{9-x^2} requires −3≤x≤3-3\le x\le3.

The symbols x∈Rx\in\mathbb R mean that xx belongs to the real numbers. An interval such as [2,5)[2,5) includes 22 and excludes 55; it means 2≤x<52\le x<5.

Worked example 1: a square-root endpoint

Question. Find the largest real domain of f(x)=x−2f(x)=\sqrt{x-2}.

Plan. Require the expression under the root to be nonnegative.

x−2≥0⇒x≥2.x-2\ge0\quad\Rightarrow\quad x\ge2.

Answer. x∈[2,∞)x\in[2,\infty), or x≥2x\ge2.

Check. At the boundary, f(2)=0f(2)=0, which exists. An input such as x=1x=1 would require the non-real value −1\sqrt{-1}.

Change only the position of the same square root: put it in a denominator. The boundary is now forbidden.

Worked example 2: excluding a zero denominator

Question. Find the largest real domain of g(x)=1x−2g(x)=\dfrac1{\sqrt{x-2}}.

Plan. Require the root to exist and also be nonzero.

x−2≥0andx−2≠0⇒x>2.x-2\ge0\quad\text{and}\quad\sqrt{x-2}\ne0 \quad\Rightarrow\quad x>2.

Answer. x∈(2,∞)x\in(2,\infty), or x>2x>2.

Check. g(3)=1g(3)=1 exists, but g(2)=1/0g(2)=1/0 does not. The strict inequality captures exactly this difference from Example 1.

When several restrictions occur, all must hold together. For x/(x−4)\sqrt{x}/(x-4), the domain is x≥0x\ge0, with x≠4x\ne4. Cancelling a factor in a rational rule does not remove an original exclusion.

Watch out

A square root permits zero; a denominator does not. Check where the root appears before deciding whether the boundary uses ≥\ge or >>.

Apply both conditions to the same expression.

Finding a range

The range records every output reached from the stated domain, including whether a limiting value is actually attained.

For q(x)=(x−2)2+1q(x)=(x-2)^2+1 on all real inputs, the square is at least zero, so q(x)≥1q(x)\ge1. Every value at least 11 occurs. On the restricted domain 3≤x≤53\le x\le5, the square runs from 11 to 99, giving 2≤q(x)≤102\le q(x)\le10. A range therefore belongs to a rule and its domain.

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