MATH01 CO1 Lesson 2 Inverse Functions PDF

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MATHEMATICS and ABM CLUSTER SCHOOL YEAR 2021 - 2022

General Mathematics MATH01

Course Outcome 1 Quarter 01

Prepared by:

Hurtado, Prince Jude M.

MATH01 | Core | CO1 E-mail address: [email protected] Telephone number: (02) 8247 - 5000

MATH01 | General Mathematics

NAME: ____________________________________ YEAR AND SECTION: _______________________

Lesson 2

TEACHER: ________________________ SCHEDULE: _______________________

Inverse Function COURSE OUTCOME BULLETIN

Objective: Illustrate, evaluate, and solve problems including inverse functions. Subject Matters: Lesson 2. Inverse Function Lesson 2.1. One-to-One Functios Lesson 2.2. Representation of Inverse Functions Lesson 2.3. Domain and Range of Inverse Functions Lesson 2.4. Application of Inverse Functions

Learning Competencies: The learner… I. Determines the inverse of a one-to-one function. II. Represents an inverse function through its: (a) table of values, and (b) graph. III. Finds the domain and range of an inverse function. IV. Graphs inverse functions V. Solves problems involving inverse functions. VI. Represents real-life situations using one-to-one functions. Evaluation: Performance Task: PBA1 (to be deployed on ALEKS, week 3) Written Work: MA1: MATLAB Activity (to be deployed on week 3)

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What do I know? (Pre-test) (Reference: Verzosa, D. M. B., Ph.D. (2016). Teaching Guide for Senior High School: General Mathematics. Published by the Commission on Higher Education.)

Determine whether the given relation is a function. If it is a function, determine whether it is oneto-one. ____________ 1. The relation pairing an SSS member to his or her SSS number.

____________2. The relation pairing a real number to its square. ____________3. The relation pairing an airport to its airport code. Airport codes are three letter codes used to uniquely identify airports around the world and prominently displayed on checkedin bags to denote the destination of these bags. Here are some examples of airport codes: MNL – Ninoy Aquino International Airport (All terminals) CEB – Mactan-Cebu International Airport DVO – Francisco Bangoy International Airport (Davao) JFK – John F. Kennedy International Airport (New York City) ____________4. The relation pairing a person to his or her citizenship. ____________5. The relation pairing a distance d (in kilometers) traveled along a given jeepney route to the jeepney fare for travelling that distance.

What is it? (Reference: Algebra and Trigonometry Enhanced eText, 4th edition by Cynthia Y. Young) We defined a function as a relationship that maps an input (contained in the domain) to exactly one output (found in the range). Algebraically, each value for x can correspond to only a single value for y. Recall the square, identity, absolute value, and reciprocal functions. All of the graphs of these functions satisfy the vertical line test. Although the square function and the absolute value function map each value of x to exactly one value for y, these two functions map two values of x to the same value for y. For example, (−1, 1) and (1, 1) lie on both graphs. The identity and reciprocal functions, on the other hand, map each x to a single value for y, and no two x-values map to the same y-value. These two functions are examples of one-to-one functions. A function 𝑓(𝑥) is one-to-one if no two elements in the domain correspond to the same element in the range; that is, if 𝑥1 ≠ 𝑥2 , then 𝑓(𝑥1 ) ≠ 𝑓(𝑥2 ).

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Condition for an Inverse Function A function 𝑓 has an inverse function if and only if 𝑓 is a one-to-one function. Alternative Condition for an Inverse Function If 𝑓 is an invreasing function or a decreasing function, then 𝑓 has an inverse function.

Examples (Reference: Algebra and Trigonometry Enhanced eText, 4th edition by Cynthia Y. Young) For each of the three relations, determine whether the relation is a function. If it is a function, determine whether it is a one-to-one function.

Solution:

𝑓 = {(0, 0), (1, 1), (1, −1)} 𝑔 = {(−1, 1), (0, 0), (1, 1)} ℎ = {(−1, −1), (0, 0), (1, 1)}

Figure 1. Solution for f, g, h one-to-one functions

2.1. One-to-One Functions A function is said to have an inverse function if and only if the function is one-to-one. The function f is one-to-one if for any 𝑥1 , 𝑥2 in the domain of f, then 𝑓(𝑥1 ) ≠ 𝑓(𝑥2 ). That is, the same y-value is never paired with two different x-values.

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Examples 1. The relation pairing an SSS member to his or her SSS number. Solution: Each SSS member is assigned with a unique SSS number. Thus, the relation is a function. Furthermore, two different members cannot be assigned to the same SSS number. Thus, the function is one-to-one. 2. The relation pairing a real number to its square. Solution: Each real number has a unique perfect square. Thus, the relation is a function. However, two different real numbers such as 2 and -2 may have the same square. Thus, the function is not one-to-one.

3. The relation pairing an airport to its airport code. Airport codes are three letter codes used to uniquely identify airports around the world and prominently displayed on checked-in bags to denote the destination of these bags. Here are some examples of airport codes: * MNL – Ninoy Aquino International Airport (All terminals) * CEB – Mactan-Cebu International Airport * DVO – Francisco Bangoy International Airport (Davao) * JFK – John F. Kennedy International Airport (New York City) * CDG – Charles de Gaulle International Airport (Paris, France) Airport codes can be looked up at http://www.world-airport-codes.com

Solution: Since each airport has a unique airport code, then the relation is a function. Also, since no two airports share the same airport code, then the function is one-to-one.

4. The relation pairing a person to his or her citizenship. Solution: The relation is not a function because a person can have dual citizenship (i.e., citizenship is not unique). 5 MATH01 | Core | CO1

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5. The relation pairing a distance d (in kilometers) traveled along a given jeepney route to the jeepney fare for travelling that distance. Solution: The relation is a function since each distance traveled along a given jeepney route has an official fare. Jeepney fare may be represented by a piecewise function, as shown below: 8.00 𝐹(𝑑) = { (8.00 + 1.50⌊𝑑⌋)

Note that ⌊𝑑⌋ is the floor function applied to d.

𝑖𝑓 0 < 𝑑 ≤ 4 𝑖𝑓 𝑑 > 4

The floor function (also known as the greatest integer function) ⌊∙⌋ : R → Z of a real number x denotes the greatest integer less than or equal to x. For example: ⌊5⌋ = 5, ⌊6.359⌋ = 6, ⌊√7⌋ = 2, ⌊𝜋⌋ = 3, ⌊−13.42⌋ = −14 In general, ⌊𝑥⌋ is the unique integer satisfying ⌊𝑥⌋ ≤ 𝑥 < ⌊𝑥⌋ + 1. The graph of the floor function is discontinuous at every integer.

Figure 2.1a. Graph of floor function ⌊𝑥⌋

If the distance to be traveled is 3 kilometers, then 𝐹(3) = 8, however, the function is not oneto-one because different distances (e.g., 2, 3 or 4 kilometers) are charged the same rate (₱8.00). That is, because 𝐹(3) = 𝐹(2) = 𝐹(3.5) = 8, then F is not one-to-one.

A simple way to determine if a given graph is that of a one-to-one function is by using the Horizontal Line Test. 6 MATH01 | Core | CO1

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Horizontal Line Test A function is one-to-one if each horizontal line does not intersect the graph at more than one point. All functions satisfy the vertical line test. All one-to-one functions satisfy both the vertical and horizontal line tests.

Examples: 1.

2.

----------------------------

------------------------

Figure 2.1b. Graph showing Horizontal Line Test

Figure 2.1c. Graph showing Horizontal Line Test

3.

4.

-----------------------------------------------

------------------------

-----------------------Figure 2.1d. Graph showing Horizontal Line Test

Figure 2.1e. Graph showing Horizontal Line Test

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Solution: The second, third, and fourth graphs represent one-to-one functions. The first graph is a quadratic function. It is not one-to-one because y-value 0 is paired with two x-values, namely 2 and -2. The first graph showing the plot of 𝑦 = 𝑥 2 − 4 fails the horizontal line test because some lines intersect the graph at more than one point. The remaining three graphs pass the horizontal line test, because all horizontal lines intersect the graph at most once.

2.2. Representation of Inverse Functions If f is a one-to-one function satisfying 𝑦 = 𝑓(𝑥), then there is a function 𝑓 −1, called the inverse function of f, such that its set consists of the ordered pairs 𝑥 = 𝑓(𝑦) defined by 𝑥 = 𝑓 −1(𝑦) if and only if 𝑦 = 𝑓(𝑥) . Consequently, 𝑓 −1 is the inverse of the function if and only if 𝑓 −1 (𝑓(𝑥)) = 𝑓(𝑓 −1 (𝑥)) = 𝑥. The domain of 𝑓 −1 is the range of f and the range of 𝑓 −1 is the domain of f.

2.3. Domain and Range of Inverse Functions Every function 𝑦 = 𝑓(𝑥) has an inverse relation 𝑥 = 𝑓(𝑦).

The ordered pairs of: 𝑦 = |𝑥| + 1 𝑎𝑟𝑒 {(−2, 3), (−1, 2), (0, 1), (1, 2), (2, 3)}

𝑥 = |𝑦| + 1 𝑎𝑟𝑒 {(3, −2), (2, −1), (1, 0), (2, 1), (3, 2)} The inverse relation is not a function since it pairs 2 to both -1 and 1.

The ordered pairs of the function f are reversed to produce the ordered pairs of the inverse relation. 8 MATH01 | Core | CO1

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Example Given the function 𝑓 = {(1, 1), (2, 3), (3, 1), (4, 2)}, its domain is {1, 2, 3, 4} and its range is {1, 2, 3}. The inverse relation of f is {(1, 1), (3, 2), (1, 3), (2, 4)}. The domain of the inverse relation is the range of the original function. The range of the inverse relation is the domain of the original function. Domain of 𝑓 −1 = Range of 𝑓

Range of 𝑓 −1 = Domain of 𝑓

If 𝑓(1) = 5, 𝑓(3) = 7, and 𝑓(8) = 10, find 𝑓 −1(5), 𝑓 −1(7), 𝑎𝑛𝑑 𝑓 −1 (10). Solution: 𝑓 −1(5) = 1

𝑓 −1(7) = 3

𝑓 −1(10) = 8

2.4. Application of Inverse Functions General Steps in Finding the Inverse of a Function 1. Express the function y as a function of x. 2. Interchange the variable. Substitute 𝑦 = 𝑥 , and 𝑥 = 𝑦. 3. Solve for y in terms of x. 4. Check if it satisfies 𝑓 −1 (𝑓(𝑥)) or 𝑓(𝑓 −1 (𝑥)). Example: 1. Find the inverse of 𝑓(𝑥) = 3𝑥 + 1. Solution: Step 1:

The equation of the function is 𝑦 = 3𝑥 + 1

Step 2:

Interchange the x and y variables: 𝑥 = 3𝑦 + 1

Step 3:

Solve for y in terms of x. 𝑥 = 3𝑦 + 1 𝑥 − 1 = 3𝑦

Transpose 1 to the left side of the equation Divide both sides by 3 to isolate y

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MATH01 | General Mathematics 𝑥−1 3

=𝑦➔𝑦 =

(𝑥−1)

Switch left and right side of the equation

3

Therefore, the inverse of 𝑓(𝑥) = 3𝑥 + 1 is 𝑓 −1(𝑥) =

𝑥−1 . 3

2. Find the inverse of 𝑔(𝑥) = 𝑥 3 − 2. Solution: Step 1:

The equation of the function is 𝑦 = 𝑥 3 − 2

Step 2:

Interchange the x and y variables: 𝑥 = 𝑦 3 − 2

Step 3:

Solve for y in terms of x. 𝑥 = 𝑦3 − 2

Transpose -2 to the left side of the equation

𝑥 + 2 = 𝑦3

Take cube root of both sides of the equation

√𝑥 + 2 = 𝑦 ➔ 𝑦 = √𝑥 + 2

3

3

Switch left and right side of the equation

The Inverse of 𝑔(𝑥) = 𝑥 3 − 2 is 𝑔−1 (𝑥) = √𝑥 + 2. 3

2𝑥+1

3. Find the inverse of the rational function 𝑓(𝑥) = 3𝑥−4. Solution: 2𝑥+1

Step 1:

The equation of the function is 𝑦 =

Step 2:

Interchange the x and y variables: 𝑥 =

Step 3:

Solve for y in terms of x: 2𝑦+1

𝑥 = 3𝑦−4

3𝑥−4

2𝑦+1

3𝑦−4

Multiply both sides by 3𝑦 − 4

𝑥(3𝑦 − 4) = 2𝑦 + 1

Distribute x into 3𝑦 − 4

3𝑥𝑦 − 4𝑥 = 2𝑦 + 1

Transpose −4𝑥 to the right and 2𝑦 to the left

(Placing all terms with y on one side and those without y on the other side) 3𝑥𝑦 − 2𝑦 = 1 + 4𝑥

Factor y on the left side of the equation

𝑦(3𝑥 − 2) = 4𝑥 + 1

To isolate y, divide both sides by 3𝑥 − 2

4𝑥+1

𝑦 = 3𝑥−2

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Therefore, the inverse of 𝑓(𝑥) is 𝑓 −1 (𝑥) =

4𝑥+1

.

3𝑥−2

4. Find the inverse of 𝑓(𝑥) = 𝑥 2 + 4𝑥 − 2 , if it exists. Solution: The given function is a quadratic function with a graph in the shape of a parabola that opens upwards. It is not a one-to-one function as it fails the horizontal line test. We can show that applying the step-by-step procedure for finding the inverse to this function leads to a result which is not a function. Step 1:

The equation of the function is 𝑦 = 𝑥 2 + 4𝑥 − 2

Step 2:

Interchange the x and y variables: 𝑥 = 𝑦 2 + 4𝑦 − 2

Step 3:

Solve for y in terms of x. 𝑥 = 𝑦 2 + 4𝑦 − 2

Transpose -2 to the left (to place all terms with y on the other side and those

without y on the other side) 𝑥 + 2 = 𝑦 2 + 4𝑦 𝑥 + 2 + 4 = 𝑦 2 + 4𝑦 + 4 𝑥 + 6 = (𝑦 + 2)2 ±√𝑥 + 6 = 𝑦 + 2

Complete the square by adding 4 on both sides Simplify 2 + 4 = 6

and 𝑦 2 + 4𝑦 + 4 = (𝑦 + 2)2 perfect square Take the square root of both sides Transpose 2 to the left to isolate y

±√𝑥 + 6 − 2 = 𝑦 ➔ 𝑦 = ±√𝑥 + 6 − 2 Switch left and right side of the equation The equation 𝑦 = ±√𝑥 + 6 − 2 does not represent a function because there are some xvalues that correspond to two different y-values (e.g., if 𝑥 = 3 , y can be 1 or -5). Therefore, the function 𝑓(𝑥) = 𝑥 2 + 4𝑥 − 2 has no inverse function.

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5. Find the inverse of 𝑓(𝑥) = |3𝑥|, if it exists. Solution: Recall that the graph of 𝑦 = |3𝑥| is shaped like a “V” whose vertex is located at the origin. This function fails the horizontal line test and therefore has no inverse.

What’s more? (Reference: Algebra and Trigonometry Enhanced eText, 4th edition by Cynthia Y. Young) A. Determine whether the given relation is a function. If it is a function, determine whether it is a one-to-one function. 1.

2.

3. {(0, 1), (1, 2), (2, 3), (3, 4)}

4. {(0, 0), (9, −3), (4, −2), (4, 2), (9, 3)}

5. {(0, 1), (1, 0), (2, 1), (−2, 1), (5, 4), (−3, 4)} 12

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B. Determine algebraically whether the function is one-to-one. 1. 𝑓(𝑥) = |𝑥 − 3| 2. 𝑓(𝑥) =

1

𝑥−1

3. 𝑓(𝑥) = 𝑥 2 − 4 4. 𝑓(𝑥) = 𝑥 3 − 1

What I have learned In atleast five sentences, write down the challenges you have experienced in learning about the basic concepts of inverse functions, and what did you do to overcome these challenges? ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________

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What I can do Verify that the function 𝑓 −1(𝑥) is the inverse of 𝑓(𝑥) by showing that 𝑓(𝑓 −1 (𝑥)) = 𝑥 and 𝑓 −1(𝑓 (𝑥)) = 𝑥. Graph 𝑓(𝑥) and 𝑦 = 𝑓 −1 (𝑥) on the same axes to show the symmetry abot the line 𝑦 = 𝑥 . 1. 𝑓(𝑥) = 2𝑥 + 1; 𝑓 −1 (𝑥) = 1

𝑥−1 2

1 𝑥

2. 𝑓(𝑥) = ; 𝑓 −1 (𝑥) = , 𝑥 ≠ 0 𝑥

Assessment (Post-test) (Reference: Verzosa, D. M. B., Ph.D. (2016). Teaching Guide for Senior High School: General Mathematics. Published by the Commission on Higher Education.)

Determine whether the given relation is a function. If it is a function, determine whether it is oneto-one. ____________ 1. The relation pairing an SSS member to his or her SSS number.

____________2. The relation pairing a real number to its square. ____________3. The relation pairing an airport to its airport code. Airport codes are three letter codes used to uniquely identify airports around the world and prominently displayed on checkedin bags to denote the destination of these bags. Here are some examples of airport codes: MNL – Ninoy Aquino International Airport (All terminals) CEB – Mactan-Cebu International Airport DVO – Francisco Bangoy International Airport (Davao) JFK – John F. Kennedy International Airport (New York City) ____________4. The relation pairing a person to his or her citizenship. ____________5. The relation pairing a distance d (in kilometers) traveled along a given jeepney route to the jeepney fare for travelling that distance.

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Answer Key

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References Young, C. Y. (2017). Algebra and Trigonometry, Enhanced eText, 4th Edition. John Wiley & Sons, Inc. Verzosa, D. M. B., Ph.D. (2016). Teaching Guide for Senior High School: General Mathematics. Published by the Commission on Higher Education. http://mathonweb.com/help_ebook/html/functions_4.htm Types of Functions and Graphs. http://www.purplemath.com/modules/fcns2.htm Domain and Range. https://brilliant.org/wiki/floorfunction/#:~:text=Floor%20Function,than%20or%20equal%20to%20x. Floor Function (Greatest Integer Function).

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