Chapter 3: The Inverse Function SHMth1: General Mathematics Accountancy, Business and Management (ABM) Mr. Migo M. Mendoza
Chapter 3: The Inverse Function Lecture 11: The One-to-One Correspondence or The One-to-One (Bijective) Function Lecture 12: The Inverse of a One-to- One Correspondence or a Bijective Function Lecture 13: The Inverse Function
Lecture 11: The One-to-One Correspondence or The One-to- One (Bijective) Function SHMth1: General Mathematics Accountancy, Business and Management (ABM) Mr. Migo M. Mendoza
A Short Recap: What do you still remember about a one-to-one correspondence/ function or a bijective function we have discussed in Chapter 1?
The One-to-One Correspondence or the Bijective Function When a function has additional property that no two unique elements of the domain have the same image in the range, the function is said to be ONE-TO-ONE. This suggests that each element of the range is the image of exactly one element of the domain.
Theorem 3.1: One-to-One Function Theorem A function that is increasing or decreasing over its domain is a oneto-one function.
Classroom Task: Identify which of the following show/s a concept of one-to-one relationship or BIJECTIVE.
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Real-Life Situation Number 2: DepEd is developing a system of identification for all learners of Philippine public schools. This is the Learner's Identification Number System (LIS) that aims to provide a unique LIS to every public school learner. Its aim is that no two LIS is assigned to a Filipino learner, and that no two Filipino learners have the same LIS.
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A Short Recap How can we determine if a given function is one-to-one or BIJECTIVE?
Theorem 3.2: The Horizontal Line Test Theorem If every horizontal line intersects the graph of function f in at most one point, then function f is one-to-one.
Classroom Task: A Short Recap: Determine if the given below is a one-to-one function. domain = { a, b, c, d, e, } codomain = {1, 2, 3, 4, 5, } f = { (a, 2), (b, 3), (c, 1), (d, 5), (e, 4)}
Final Answer: Using the ellipse diagram or arrow diagram, we have proven that f = { (a, 2), (b, 3), (c, 1), (d, 5), (e, 4)} is a one-to-one function.
Classroom Task: A Short Recap: Determine if the given below is a one-to-one function. domain = { m, i, g, 0, } codomain = {0, 7, 5 } f = { (m, 0), (i, 5), (g, 5), (o, 7) }
Final Answer: Using the ellipse diagram or arrow diagram, we have proven that f = { (m, 0), (i, 5), (g, 5), (o, 7) } is not a one-to-one function but onto or surjective.
Classroom Task: A Short Recap: Determine if the given below is a one-to-one function. domain = { D, L, S, U} codomain = {0, 7, 5, 2, 1, 6, } f = { (L, 0), (D, 7), (S, 1), (U, 5) }
Final Answer: Using the ellipse diagram or arrow diagram, we have proven that f = { (L, 0), (D, 7), (S, 1), (U, 5) } is a NOT A ONE-TO-ONE FUNCTION but INJECTIVE.
Lecture 12: The Inverse of a Oneto-One Correspondence or Bijective Function SHMth1: General Mathematics Accountancy, Business and Management (ABM) Mr. Migo M. Mendoza
Something to think about Reverse the domain and the range of each of the given functions on the board. Then, identify if its reverse is a function. Afterwards, use this classroom task to construct your own definition of an INVERSE FUNCTION.
Example 60: Determine if the given below is a one-to-one function. domain = {1, 2, 3, 4, 5, } codomain = { a, b, c, d, e, } f -1 = { (2, a), (3, b), (1, c), (5, d), (4, e)}
Final Answer: Using the ellipse diagram or arrow diagram, we have proven that f -1 = { (2, a), (3, b), (1, c), (5, d), (4, e)} is a FUNCTION and a one-to-one or BIJECTIVE.
Example 61: Determine if the given below is a one-to-one function. domain = {0, 7, 5 } codomain = { m, i, g, 0, } f -1 = { (0, m), (5, i), (5, g), (7, o) }
Final Answer: Using the ellipse diagram or arrow diagram, we have proven that f -1 = { (m, 0), (i, 5), (g, 5), (o, 7) } is NOT A FUNCTION.
Example 62: Determine if the given below is a one-to-one function. domain = {0, 7, 5, 2, 1, 6, } codomain = { D, L, S, U} f -1 = { (0, L), (7, D), (1, S), (5, U) }
Final Answer: Using the ellipse diagram or arrow diagram, we have proven that f -1 = { (0, L), (7, D), (1, S), (5, U) } is NOT A FUNCTION.
Something to think about Based on our previous classroom task, what insight(s) have you learned which you can use to construct your own definition of an INVERSE FUNCTION?
Definition of an Inverse Function The inverse function or inverse of a function is a set of ordered pairs formed by reversing the coordinates of ordered pair of the function.
The Domain and the Range of the Inverse Function The domain of the inverse function is the range of the function, and the range of the inverse function is the domain of the function.
The Symbols Domain of f =Range of f -1 Range of f=domain of f -1
Something to think about Based on our previous examples, when can we say that a function has an inverse function? Or when can we say that the inverse of a function is a function?
Take Note: A function f has an inverse if and only if f is a one-to-one correspondence or a BIJECTIVE FUNCTION.
Take Note: Not all function has an INVERSE.
Performance Task 10: Please download, print and answer the Let s Practice 10. Kindly work independently.
Lecture 13: Finding Inverse of an Equation SHMth1: General Mathematics Accountancy, Business and Management (ABM) Mr. Migo M. Mendoza
Example 63: The function f is one-to-one. Find the inverse and check the answer. f ( x) 2x 4
Steps in Finding the Inverse of an One-to-One Equation: Step 1: Replace f(x) by y.
Steps in Finding the Inverse of an One-to-One Equation: Step 2: Interchange x and y.
Steps in Finding the Inverse of an One-to-One Equation: Step 3: Solve for y in terms of x.
Steps in Finding the Inverse of an One-to-One Equation: Step 4: Replace y with f -1 (x).
Steps in Finding the Inverse of an One-to-One Equation: Step 5: Verify if f(x) and f -1 (x) are inverses of each other.
Something to think about What previous lesson can we apply in order to verify if f(x) and f -1 (x) are inverses of each other?
The Composition of an Inverse Function The composition of an inverse function states that if the inverse relation of a function f is also a function, it is called the inverse function of f, denoted as f -1.
Moreover: A function and its inverse are related by the following equations: 1 f f ( x) x for all values of x in the domain of f -1 ; and 1 f f ( x) x for all values of x in the domain of f. Thus: f 1 1 f ( x) f f ( x) x.
Take Note: Take note that f -1 (x) is not the reciprocal of f(x) but the notation for the inverse of a one-to-one function.
Final Answer: The inverse of the function f (x) is: f 1 ( x) 1 x 2 2
Take Note: If two functions are inverses of each other, then their graphs are mirror images with respect to the graph of the line y x.
Example 64: Graph f ( x) 2x 4 and f 1 ( x) 1 x 2 to show that their 2 graphs are symmetric with respect to the graph of the line y = x.
Example 65: The function f is one-to-one. Find the inverse and check the answer. f ( x) 2 x 7
Final Answer: The inverse of the function f (x) is: f 7x 1 ( x) x 2
Example 66: f Determine whether ( x) 2x 3 and are inverses of each other. g( x) x 2 3
Final Answer: Since, f [ g( x)] x and g[ f ( x)] x, f(x) and g(x) are inverses of each other.
Example 67: f Verify whether ( x) 3x 3 and are inverses of each other. g( x) x 3 2
Final Answer: Since, f [ g( x)] x and g[ f ( x)] x, f(x) and g(x) are NOT inverses of each other.
Performance Task 11: Please download, print and answer the Let s Practice 11. Kindly work independently.