Mathematics
Grade 11
15 min
Identify functions
Identify functions
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1
Introduction & Learning Objectives
Learning Objectives
Define a function as a special type of relation with a unique output for each input.
Distinguish between a function and a relation using sets of ordered pairs, mapping diagrams, and tables.
Apply the Vertical Line Test to determine if a graph represents a function.
Algebraically determine if an equation in two variables represents a function of x.
Identify the domain and range of a given relation to support the identification of a function.
Articulate why a given relation is or is not a function, providing specific evidence.
Think of your student ID number. Can one ID number belong to two different students? 🤔 The predictable, one-to-one nature of that relationship is the core idea of a function!
This tutorial will establish the fundamental definition o...
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Key Concepts & Vocabulary
TermDefinitionExample
RelationAny set of ordered pairs, (x, y), which links elements from a set of inputs to a set of outputs.The set R = {(1, 2), (2, 4), (1, 5)} is a relation.
FunctionA special type of relation where every input (x-value) is paired with exactly one output (y-value). No input can have more than one output.The set F = {(1, 2), (2, 4), (3, 6)} is a function because each x-value (1, 2, 3) has only one corresponding y-value.
DomainThe set of all possible input values (the x-coordinates) in a relation.For the relation R = {(1, 2), (2, 4), (1, 5)}, the domain is {1, 2}.
RangeThe set of all possible output values (the y-coordinates) in a relation.For the relation R = {(1, 2), (2, 4), (1, 5)}, the range is {2, 4, 5}.
Mapping DiagramA visual tool that shows how each element of th...
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Core Formulas
The Uniqueness Rule
A relation is a function if for any input x in the domain, there is one and only one output y in the range. If (x, y₁) and (x, y₂) are in the relation, then it must be that y₁ = y₂.
This is the fundamental definition of a function. Use it when analyzing sets of ordered pairs, tables, or mapping diagrams. If you find a single x-value paired with two or more different y-values, the relation is not a function.
The Vertical Line Test
A graph in the Cartesian plane represents a function if and only if no vertical line intersects the graph at more than one point.
This is a graphical method for identifying functions. If you can draw even one vertical line, x = a, that crosses the graph in two or more places, the graph does not represent a function.
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Challenging
A relation is described as 'y is a square root of x'. Is this relation a function for x > 0?
A.Yes, because the principal square root is always used.
B.No, because every positive number x has two square roots, a positive and a negative one.
C.Yes, because the domain is restricted to positive numbers.
D.No, because the range includes negative numbers.
Challenging
Consider the piecewise-defined relation: y = { x + 2, if x < 1; and 2x + 1, if x ≥ 1 }. Is this relation a function?
A.Yes, because the value at the boundary point x=1 is uniquely defined as y=3 by the second piece.
B.No, because at the boundary point x=1, y could be 3 from the first piece or 3 from the second piece.
C.No, because there are two different rules for y.
D.Yes, because both x+2 and 2x+1 are functions on their own.
Challenging
Does the equation x = y⁴ - 2 represent y as a function of x?
A.Yes, because it is a polynomial.
B.No, because y is raised to an even power, leading to two possible y-values for a given x.
C.Yes, because if you solve for y, you get a single expression.
D.No, because the domain is restricted.
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