U1 L1.1: Representing Real-Life Situations using Functions

• A relation is a rule that relates an element of one set to one or more elements in another set
• In a function, the input has exactly one output, creating a many-to-one correspondence where many inputs are mapped to one element
• In an ordered pair x, y, x is the input and y is the output generated by a machine depending on some rule
• In a relation, elements from set A are mapped to elements in set B using some rule f
• Before starting this lesson, it is suggested to revisit the following topics from previous math lessons:
• A relation is a set of ordered pairs x, y where x is an element of the domain of the relation, and y is an element of the range of the relation, related by some rule
• An input can have more than one output, or different inputs can produce the same output
• A function is a relation that relates an element of one set to exactly one element in another set by some rule
• One-to-one correspondence


  Each element in S is mapped to exactly one element in T
• One-to-many correspondence

  One input is mapped to multiple outputs
• For a relation to be considered as a function, each input (x) should have exactly one output (y)
• We defined a relation earlier as a set of ordered pairs x, y. How do we know whether it is a function or not?
• Consider the relation f = 1,2 , 3,7 , 5,12 , 7,17. The relation f is a function as each x-value has exactly one y-value related to it
• The relation in Figure 1.1.3 is not considered as a function. The inputs 1 and 3 have more than one output
• Many-to-one correspondence


  Many inputs are mapped to one element
• A set of ordered pairs x, y can be considered a function if there is no repetition in the x-values, implying each x-value is related to only one y-value
• Function since each input has exactly one output
• Example 1: Determine if the following relations are functions
• Solution: a. E = 0,1 , 1,4 , 2,7 , 3,10 The relation E is a function since no x-values are repeated implying that each x-value is related to only one y-value
• The relation in Figure 1.1.2 is considered to be a function since each element in S is mapped to exactly one element in T
• Consider another relation g = 2, -3 , 0,2 , -2, -1 , 2,3. The relation g is not a function as x = 2 is related to two y-values, -3 and 3
• Relations


  • E = 0,1 , 1,4 , 2,7 , 3,10
  • F = 2,4 , -3,9 , 0,0 , -1,1
  • G = C D , 3 , -2,0 , 0,4 , 1, -6 , -2,8
• A relation is considered as a function if it is one-to-one or many-to-one
• Relation is a function if each 𝑥-value is related to only one 𝑦-value
• Relation 𝐹 is a function since no 𝑥-values are repeated, implying that each 𝑥-value is related to only one 𝑦-value
• Definition of Vertical Line Test: If a vertical line intersects the graph of a relation at exactly one point, then the graph represents a function. Otherwise, the graph is not a function
• The graph of a relation where a vertical line intersects at two points implies that the 𝑥-value has two 𝑦-values related to it, making it not a function
• Relation 𝐺 is not a function since 𝑥 = 2 is repeated in the relation, implying that this 𝑥-value is related to more than one 𝑦-value; thus, violating the definition of a function
• Equations can describe functions and relations
• The graph that intersects the vertical line test at exactly one point represents a function
• The equation 𝟐𝒙 − 𝒚 = 𝟑 is not specified as a function or not
• The equation 𝑦 = 4 − 𝑥D is not a function as it generates two 𝑦-values for �� = 0
• If there exists an 𝑥-value that generates more than one 𝑦-value, then the equation is not a function
• The relation 𝑦 = 𝑥F + 2 is considered a function as each 𝑥-value has only one 𝑦-value
• The vertical line test can be drawn anywhere that will intersect the graph
• Functions can represent real-life situations
• The equation 𝑥 + 𝑦 = 1 is a function as no x-values generate more than one y-value
• The given graph of the relation is not a function
• Examples of equations as functions


  • 𝑦 = 𝑥D + 1
  • 𝑥D + 𝑦D = 2
  • 𝑦 = 𝑥 + 1
• 𝟐𝒙 − 𝒚 = 𝟑 is a function