Systems of Linear and Inequalities & RELATION AND FUNCTION

Systems of Linear and Inequalities & RELATION AND FUNCTION

By James Clark

RELATION AND FUNCTION

​In this lesson, you will be able to learn the following:

illustrate a relation and a function;

verify if a given relation is a function;

determine dependent and independent variables;

illustrate and graph a linear function;

solve problems involving linear function; and

find the domain and range of a function.​

​A relation is a set of ordered pairs. The domain of a relation is the set of first coordinates. The range is the set of second coordinates. A function is a relation in which element of the domain corresponds to exactly one element of the range. The members of the domain can be called inputs, and the members of the range can be called outputs when a number is compared to a number-processing machine. Shown below is one of a real-life situation where the value of one variable depends on the value of another variable.

​A supermarket holds a closing-out sale. All merchandise is sold at a 30% discount. For the convenience of the shoppers, the marketing supervisor considers a table of marked prices (x) and their corresponding selling prices (y).

​For each input (value of x), the number-processing machine yields an output (value of y). For example when the input values are x = 50, 100, and 150, the output values are y = 35, 70, 105, respectively. Because your input an x-value to represent the marked price, x is called the independent variable. Since the selling price depends on the marked price, y is called the dependent variable.

• Independent Variable – the input in a function.

• Dependent Variable – the output in a function

​A function can be presented in different ways: a. A table of values List of the input values in one row or column and output values in another row or column.

Marked Price (x)​	​50	​100	​150	​200	​250	​300	​350
Selling Price (y)ccc	​35	75	​105	​140	​175	​210	​245

​b. Ordered pairs Write inputs as first coordinates and the outputs as the second coordinates. (50, 35), (100, 70), (150, 105), (200, 140), (250, 175), (300, 210), (350, 245)

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​c. A graph

Plot the ordered pairs.

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​300

250

200

150

100

50

0

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​Selling Price (y)

​	​	​	​
​	​	​	​
​	​	​	​
​	​	​	​
​	​	​	​
​	​	​	​

​0

​100

​300

​200

​400

​The Vertical Line Test

A graph is a function if you draw a vertical line on the plan, move it from left to right, and touch the graph at just one point.

​Marked Price (x)

​d. An equation y = x – 0.3x or y = 0.7x, where x is the independent variable and y is the dependent variable.

A function of the form f(x) = mx + b, where m ≠ 0 is called linear function.

​Linear Functions, Equations and Graphs

​RELATION AND FUNCTION

Example: Determine whether the following relations are functions or not.

​a. {(1, 4), (2, 5), (3, 6), (4, 7)} c. 𝑦 2+ 1 = x

b.

Input	Output​
​1	​2
​2	​4
​3	​8

​Solution:

a. Each element in the domain {1, 2, 3, 4} is assigned no more than one value in the range; 1 is assigned only to 4; 2 is assigned only to 5; 3 is assigned only to 6; 4 is assigned only to 7. Therefore, it is a function.

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b. Input: 1 2 3 Output: 2 4 8 Each input value matches with only one output value. So this relationship is a function. The domain is {1, 2, 3} and the range is {2, 4, 8}

​

c. Solve for y:

𝑦 2+ 1 = x

𝑦 2 = x – 1

y = ±√𝑥 − 1

​

This means that for every value of x, there correspond to two values for y. Therefore, y is not a function of x.

​Here are the Characteristics of a Function:

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1. Each element in domain X must be matched with exactly one element in range Y.

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2. Some elements in Y may not be matched with any element in X.

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3. Two or more elements in X may be matched with the same element in Y.

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​GRAPH OF A LINEAR FUNCTION

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We will follow the same rule as we graph linear equations given on the previous topics to graph a linear function

Domain and Range, Intercepts, the slope of a line​

THE DOMAIN AND RANGE OF A RELATION​

In an equation or inequality, the domain of a relation is the set of all permissible values of x that keep y real, In contrast, the range of a relation is the set of all permissible values of y that keep x real.​

​

Example: Find the domain and range of each of the following:

a.​

Input​	​1	​2	​3
​	​2	​4	​8

SOLUTION:

Domain: the elements in the input Thus, the domain is {1, 2, 3}. Range: the elements in the output. Thus, the range is {2, 4, 8}.​

​b. {(1, 4), (2, 5), (3, 6), (4, 7)} SOLUTION: Domain: the elements in the input Thus, the domain is {1, 2, 3, 4}. Range: the elements in the output. Thus, the range is {4, 5, 6, 7}.

c. 𝑥 2 + y – 3 = 0 SOLUTION: To find the domain, solve for y: 𝑥 2 + y – 3 = 0 y = 3 – 𝑥 2​

​Domain: the set of real numbers or D = {x | ∈ R}.

Solve for x to find the range

​𝑥 2 + y – 3 = 0

𝑥 2 = 3 – y

x = ± √3 − y

​For x to be a real number, the radicand 3 – y must be non-negative. That is,

3 – y ≥ 0 or y ≤ 3.​

​Thus, the range is the set of real number less than or equal to 3. In symbols, we write ​Range = {y |𝑦 ≤ 3}.

​d. 3𝑥+2 / 𝑥−3 = y

​SOLUTION: For y to be defined, the denominator x – 3 must not be equal to zero

​x – 3 ≠ 0

x ≠ 3

Thus, Domain: All real numbers not equal to 3. Solve for x to find the range.​

​𝟑𝒙+𝟐 / 𝒙−𝟑 = y x = −3𝑦−2 / 3−y

​

3x + 2 = xy – 3y​

3x – xy = - 3y -2

x(3 – y) = - 3y – 2​

​Multiply both sides by x – 3.

​Factor 3x – xy.

​3 – y ≠ 0

​y ≠ 3

​For x to be real, the denominator y – 3 must not be equal to zero.

​Thus, the range is the set of real numbers not equal to 3

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• Books

  Diaz, Zenaida B., Mojica, Maharlika P., et. Al. Next Century Mathematics 8. Phoenix Publishing House. 2014. Chapter 3. Pages 159-217.

• Mendoza, Marilyn O., Oronce, Orlando A. e-Math 8. Rex Book Store, Inc. 2013. Pages 209-215.