Discrete Mathematics PDF

UNIT ONE: SETS Definition A set is a well-defined collection of objects. 1. The collection of vowels in English alphabets 2. The collection of all districts in the Upper West Region 3. The collection of good players in the Ghana Premier League. Note this is not...

UNIT ONE: SETS Definition A set is a well-defined collection of objects. 1. The collection of vowels in English alphabets 2. The collection of all districts in the Upper West Region 3. The collection of good players in the Ghana Premier League. Note this is not a set, since the term good player is vague and is not well defined Set Notation  A common way of describing a set is to say it is the collection of all real numbers which satisfy a certain condition.  One uses this notation A = {𝑥 |𝑥 satisfies this or that condition}  The upper case letters in a calligraphic font are used to denote sets. (A, B, C, D,...) For instance, the interval (a, b) can be described as (𝑎, 𝑏) = {𝑥 |𝑎 < 𝑥 < 𝑏}  The set 𝐵 = { 𝑥 |𝑥 2 − 1 > 0} consists of all real numbers 𝑥for which 𝑥 2 − 1 > 0, i.e. it consists of all real numbers 𝑥 for which either 𝑥 > 1 or 𝑥 < −1 holds. This set consists of two parts: the interval(−∞, 1) (1, ∞)and the interval (1,∞). Note: A set is often described in the following two ways: 1. Roster Method (Tabular Form) In this method, a set is described by listing elements, separated by commas, within braces { }. Example: i. The set of vowels of the English alphabets {a, e, i, o, u} ii. The set of all prime numbers less than 11 {2, 3, 5, 7} 1  2. Rule Method (set builder form)  A set is described by characterizing property 𝑃(𝑥) of its elements 𝑋  𝑋 in such a case, the set is described by {X: P(X)holds} or {X/P(X)holds}. Example {𝑥: 𝑥 is a prime number less than 20}, {𝑥: − 2 < 𝑥 < 12} Types of Sets 1. Universal Sets A set that contains all sets in a given context is called universal set. Example, if the universal set contains the sets 𝐴 = {1, 2, 3}, 𝐵 = {2, 4, 5, 6} and 𝐶 = {1, 3, 5, 7}. Then the universal set can be described as 𝑈 = {1, 2, 3, 4, 5, 6, 7} 2. Empty set: A set is said to be empty or null or void set if it has no elements and is denoted by { } or ∅. 3. Finite set: A set is called finite set if it is either void or its elements can be listed by and the process terminates at a certain point, and the opposite is infinite set. Example: i. Set of even numbers less than 10 ii. Set of all students in level 100 at KUC iii. Set of counting numbers 4. Subsets: Let 𝐴 and 𝐵 be any two sets. If every element of 𝐴 is an element of 𝐵, then 𝐴 is a subset of 𝐵 and is denoted by 𝐴 ⊂ 𝐵. 2  5. Power Set Let 𝐴 be a set. Then the collection of family of all subsets of 𝐴 is called the power set of 𝐴 and is denoted by 𝑃(𝐴) = 2𝑛 , where 𝑛 is the number of elements in 𝐴. Note: a. Every set is a subset of itself b. The empty set is a subset of every set Operation of Sets 1. Union of sets Let 𝐴 and 𝐵 be two sets. The union of 𝐴 and 𝐵 is the set of all those elements which belong either to 𝐴 or 𝐵 or to both 𝐴 and 𝐵 and is denoted by 𝐴 ∪ 𝐵. Example, If 𝐴 = {1, 2, 3, 4, 5} and 𝐵 = {1, 3, 9, 12}, then 𝐴 ∪ 𝐵 = {1, 2, 3, 4, 5, 9, 12} 2. Intersection of sets Let 𝐴 and 𝐵 be two sets. The intersection of 𝐴 and 𝐵 is the set of all those elements that belong to both 𝐴 and 𝐵 and is denoted by 𝐴 ∩ 𝐵. Example If 𝐴 = {1, 2, 3, 4, 5} and 𝐵 = {1, 3, 9, 12}, then 𝐴 ∩ 𝐵 = {1, 3} 3. Difference of sets Let 𝐴 and 𝐵be two sets. The difference of 𝐴 and 𝐵 is the set of all those elements of 𝐴 which do not belong 𝐵and is denoted by 𝐴 − 𝐵. Example If 𝐴 = {1, 2, 3, 4, 5} and 𝐵 = {1, 3, 9, 12}, then 𝐴 − 𝐵 = {2, 4, 5} and 𝐵 − 𝐴 = {9, 12} 4. Complement of a set Let 𝑈 be the universal set and let 𝐴 be a set such that 𝐴 is a subset of 𝑈. Then the complement of 𝐴 with respect to 𝑈 is denoted by 𝐴′ 𝑜𝑟 𝑈 − 𝐴. 3 Laws of Algebra of Sets 1. Distributive laws i. 𝐴 ∪ (𝐵 ∩ 𝐶 ) = (𝐴 ∪ 𝐵 ) ∩ (𝐴 ∪ 𝐶 ) ii. 𝐴 ∩ (𝐵 ∪ 𝐶 ) = (𝐴 ∩ 𝐵 ) ∪ (𝐴 ∩ 𝐶 ) 2. De-Morgan law i. (𝐴 ∪ 𝐵 )′ = 𝐴′ ∩ 𝐵 ′ Proof LHS: Let 𝑥 ∈ (𝐴 ∪ 𝐵 )′ ⇒ 𝑥 ∉ (𝐴 ∪ 𝐵 ) ⇒ 𝑥 ∉ A and 𝑥 ∉ 𝐵 ⇒ 𝑥 ∈ 𝐴′ and 𝑥 ∈ 𝐵 ′ ⇒ 𝑥 ∈ 𝐴′ ∩ 𝐵 ′ ⇒ (𝐴 ∪ 𝐵 )′ ⊂ 𝐴′ ∩ 𝐵 ′ RHS: Now let 𝑦 ∈ 𝐴′ ∩ 𝐵 ′ ⇒ 𝑦 ∈ 𝐴′ and 𝑦 ∈ 𝐵 ′ ⇒ 𝑦 ∉ A and 𝑦 ∉ 𝐵 ⇒ 𝑦 ∉ (𝐴 ∪ 𝐵 ) ⇒ 𝑦 ∈ (𝐴 ∪ 𝐵 )′ ⇒ 𝐴′ ∩ 𝐵 ′ ⊂ (𝐴 ∪ 𝐵 )′ ii. (𝐴 ∩ 𝐵 )′ = 𝐴′ ∪ 𝐵 ′ (Try proving this) Note: If A, B and C are finite sets and U the universal set, then a. 𝑛(𝐴 ∪ 𝐵 ) = 𝑛(𝐴) + 𝑛(𝐵 ) − 𝑛(𝐴 ∩ 𝐵 ) b. 𝑛(𝑈) = 𝑛(𝐴 ∪ 𝐵 ) + 𝑛(𝐴′ ∩ 𝐵 ′ ) c. 𝑛(𝑈) = 𝑛(𝐴 ∪ 𝐵 ), 𝑖𝑓 (𝐴 ∩ 𝐵 )′ = ∅ 4 Ordered Pairs When listing the elements of a set, the order in which the elements are written is immaterial. Thus, for example , {1, 2, 3} = {2, 3, 1} = {3, 1, 2}. Often, however, we need to be able to distinguish the order in which two elements are listed. In an ordered pair of elements 𝒂 and 𝒃, denoted by (𝒂, 𝒃), the order in which the entries are written is taken into account. Thus, (𝟏, 𝟐) ≠ (𝟐, 𝟏), and (𝒂, 𝒃) = (𝒄, 𝒅) if and only if 𝒂 = 𝒄 and 𝒃 = 𝒅. Cartesian Product  Given sets 𝐴 and 𝐵, the Cartesian product of 𝐴 and 𝐵 is the set consisting of all the ordered pairs (𝒂, 𝒃), where 𝒂 ∈ 𝑨 and 𝒃 ∈ 𝑩.  The Cartesian product of A and B is denoted A x B.  Thus 𝑨 𝒙 𝑩 = {(𝒂, 𝒃): 𝒂 ∈ 𝑨 and 𝒃 ∈ 𝑩} Example: Let 𝐴 = {1, 2, 3} and 𝐵 = {3, 4}, then 𝐴 𝑥 𝐵 = {(1, 3), (1, 4), (2, 3), (2, 4), (3, 3), (3, 4), } and 𝐵 𝑥 𝐴 = {(3, 1), (3, 2), (3, 3), (4, 1), (4, 2), (4, 3), } From the results, we notice that 𝑨 𝒙 𝑩 ≠ 𝑩 𝒙 𝑨 Equivalence Relations Generally, we define a relation from a set A to a set B to be any subset of the Cartesian product 𝑨 𝒙 𝑩. If R is a relation from set A to set B and (𝒙, 𝒚)is an element of R, we will say that x is related to y by R and write 𝒙 𝑹 𝒚 instead of (𝒙, 𝒚) ∈ 𝑹. A relation from a set S to itself, that is, a subset of 𝑺 𝒙 𝑺, is called a relation on 𝑺. 5 Note: A relation R on a set S may have any of the following special properties;  Reflexive property: If for each x in S, 𝒙 𝑹 𝒙 is true, then R is reflexive  Symmetric property: If 𝒚 𝑹 𝒙 is true whenever, 𝒙 𝑹 𝒚 is true, then R is called Symmetric  Transitive property: If 𝒙 𝑹 𝒛 is true whenever 𝒙 𝑹 𝒚 and 𝒚 𝑹 𝒛 are both true, then R is called Transitive A relation that is reflexive, Symmetric, and Transitive is called an equivalence relation. Example Let S be the set of positive integers. Define a relation R on S by letting x R y mean that x divides y. Determine whether or not R is an equivalence relation on the set S. For R to be an equivalence relation, it must be reflexive, symmetric and transitive.  R is reflexive, since every positive integer divides itself  R is transitive, since if x divides y and y divides z, then x divides z.  R is not symmetric since 2𝑅8 ≠ 8𝑅2 Hence the relation R is not an equivalence relation. 6  UNIT TWO: FUNCTIONS Introduction If X and Y are two sets, a function f from X to Y is a relation from X to Y with the property that, for each element x in X, there is exactly one element y in Y such that x f y. Note that because a relation from X to Y is simply a subset of X x Y, a function is a subset S of X x Y such that for each 𝒙 ∈ 𝑿 there is a unique 𝒚 ∈ 𝒀 with (𝒙, 𝒚) in S Definition A function 𝑓 is a correspondence between two sets 𝑋 and 𝑌 that assigns to each element 𝑥 of the set 𝑋 one and only one element 𝑦 of set 𝑌. The element y being assigned is called the image of 𝑥 the set 𝑋 is called the domain of the function and the set of all images is called the range of the function X Y a f(a) b f(b) c f(c) x f(x) Domain Range The assignments made by a function are often expressed as ordered pairs. For example the assignment in the diagram could be expressed as: (𝑎, 𝑓(𝑎)), (𝑏, 𝑓(𝑏)), (𝑐, 𝑓(𝑐)), (𝑥, 𝑓(𝑥)), where the first components are from the domain and the second components from the range. Thus, a function can also be thought of as a set of ordered pairs for which no two of the ordered pairs have the same first component. A function could also be defined as a relation in which no two of the ordered pairs have the same first component. 7 To specify a function 𝑓 you must give a rule which tells you how to compute the value 𝑓(𝑥) of the function for a given real number 𝑥, and say for which real numbers 𝑥 the rule may be applied. The rule must be unambiguous: the same 𝑥 must always lead to the same 𝑓(𝑥 ).  The set of numbers for which a function is defined is called its domain.  The set of all possible numbers 𝑓(𝑥) as 𝑥 runs over the domain is called the range of the function.  The rule which specifies a function can come in many different forms. Most often it is a formula, as in 𝑓(𝑥) = 2𝑥 − 3 𝑓𝑜𝑟 𝑥  Sometimes you need a few formulas, as in 3𝑥 − 1 𝑖𝑓 𝑥 > 3 2 𝑓(𝑥 ) = {𝑥 − 2 𝑖𝑓 − 2 ≤ 𝑥 ≤ 3 2𝑥 + 3 𝑖𝑓 𝑥 < −2  Functions which are defined by different formulas on different intervals are sometimes called piecewise defined functions. The Vertical Line Property. Generally speaking, graphs of functions are curves in the plane but they distinguish themselves from arbitrary curves by the way they intersect vertical lines. The graph of a function cannot intersect a vertical line “𝑥 = constant” in more than one point. This is termed the vertical line property. The reason why this is true is very simple. If two points lie on a vertical line, then they have the same 𝑥 coordinate, so if they also lie on the graph of a function 𝑓, then their 𝑦- coordinates must also be equal, namely 𝑓 (𝑥 ). 8 Note The graph of 𝑓(𝑥 ) = 𝑥 3 − 𝑥 “goes up and down,” and, even though it intersects several horizontal lines in more than one point, it intersects every vertical line in exactly one point. The collection of points determined by the equation 𝑥 2 + 𝑦 2 = 1 is a circle. It is not the graph of a function since the vertical line 𝑥 = 0 (the y-axis) intersects the graph in two points. 3 y= x − x  The graph of 𝑦 = 𝑥 3 − 𝑥 fails the “horizontal line test,” but passes the “vertical line test.” However, the circle fails both tests. The Domain of a Function The domain of 𝑓 consists of all 𝑥 values at which the function is defined, and the range consists of all possible values 𝑓 can have. A systematic way of finding the domain and range of a function for which you are only given a formula is as follows:  The domain of 𝑓 consists of all 𝑥 for which 𝑓(𝑥 ) is well-defined;  The range of 𝑓 consists of all 𝑦 for which you can solve the equation 𝑓(𝑥 ) = 𝑦. 1 The expression 𝑥2 can be computed for all real numbers 𝑥 except 𝑥 = 0 since this leads to 1 division by zero. Hence the domain of the function 𝑓(𝑥 ) = 𝑥 2 is “all real numbers except 0” or {𝑥 |𝑥 ≠ 0} = (−∞, 0) ∪ (0, ∞) To find the range the question ask is “for which y can the equation 𝑦 = 𝑓 (𝑥 ) be solved for x?” 1 1 for instance, if 𝑦 = then we must have 𝑥 2 = , so first of all, since we have to divide by 𝑦, 𝑥2 𝑦 9  1 then 𝑦 can’t be zero. Furthermore, 𝑥 2 = 𝑦 says that 𝑦 must be positive. On the other hand, if 1 1 𝑦 > 0 then 𝑦 = has a solution (in fact two solutions), namely 𝑥 = ±. 𝑥2 𝑦 This shows that the range of 𝑓 is “all positive real numbers”, that is { 𝑥 |𝑥 > 0} 𝑜𝑟 (0, ∞). Graphing a Function. You get the graph of a function 𝑓 by drawing all points whose coordinates are (𝑥, 𝑦)where 𝑥 must be in the domain of 𝑓 and 𝑦 = 𝑓(𝑥 ). f y = f (x ) ( x,f (x )) x f In the case of a straight line graph, the line is the graph of 𝑓(𝑥 ) = 𝑚𝑥 + 𝑐. It intersects the 𝑦- axis at height 𝑐. Any function that can be written in the form 𝑓(𝑥 ) = 𝑚𝑥 + 𝑐 where 𝒎 and 𝒄, are real numbers, is called a linear function. m 1 P1 y1 y 1 − y0 P0 y0 x1 − x0 n x0 x1 10 Note The ratio between the amounts by which 𝑦 and 𝑥 increase as you move from one point to 𝑦1 −𝑦0 another on the line is: =𝑚 𝑥 1 −𝑥 0 Quadratic functions Any function that can be written in the form 𝑓(𝑥 ) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐, where 𝒂, 𝒃 and 𝒄, are real numbers and 𝑎 ≠ 0, is called a quadratic function. The general solution for any quadratic function is given by −𝑏 ± √ [𝑏2 − 4𝑎𝑐 ] 𝑥= 2𝑎 Note:  the graph of any quadratic function is called a parabola.  When 𝒂 > 𝟎: the parabola has a minimum  When 𝒂 < 𝟎: the parabola has a maximum 𝒃  The axis of symmetry of a parabola is given by: 𝒙 = − 𝟐𝒂 𝒃 𝟒𝒂𝒄−𝒃𝟐  The parabola has its vertex at (− 𝟐𝒂 , ) 𝟒𝒂 Example: Find two numbers whose sum is 30, such that the sum of their squares is a minimum. Let x represent one of the numbers; then 30 − 𝑥 represents the other number. By expressing the sum of their squares as a function of x, we have: 𝑓(𝑥 ) = 𝑥 2 + (30 − 𝑥 )2 = 𝑥 2 + 900 − 60𝑥 + 𝑥 2 = 2𝑥 2 − 60𝑥 + 900 This is a quadratic equation with 𝑎 = 2; 𝑏 = −60; 𝑐 = 900 Therefore, the x-value where the minimum occurs is: 𝑏 −60 − 2𝑎 = − = 15. If x is 15, then the other number, 30 − 𝑥 = 15 4 Therefore, the two numbers should both be 15 11 One-to-one Correspondence A function that is both one-to-one and onto is called a One-to-one correspondence. Note that if 𝑓: 𝑋 → 𝑌 is one-to-one correspondence, then for each 𝑦 ∈ 𝑌, there is exactly one 𝑥 ∈ 𝑋 such that 𝑦 = 𝑓(𝑥) Example: Let X be the set of real numbers. Show that the function f: X → Y defined by f(x) = 2x − 3 is one-to-one correspondence. In order to show that 𝒇 one-to-one, we must show that if 𝑓(𝑥 1 ) = 𝑓(𝑥2 ), Then 𝑥 1 = 𝑥 2 Let 𝑓 (𝑥1 ) = 𝑓(𝑥 2 ), then 2𝑥 1 − 3 = 2𝑥 2 − 3 2𝑥 1 − 3 = 2𝑥 2 − 3 ⇒ 2𝑥 1 = 2𝑥 2 ⇒ 2𝑥 1 = 2𝑥 2. Hence 𝒇 one-to-one In order to show that 𝒇 onto, we must show that if 𝒚 is an element of the codomain of 𝒇, then there is an element 𝒙 of the domain such that 𝑦 = 𝑓(𝑥 ). Since the domain and the codomain of 𝒇 are both the set of real numbers, we need to show that for any real number 𝑦, there is a real number 𝑥 such that 𝑦 = 𝑓 (𝑥 ). 𝑦+3 Take 𝑥 = , (this value was found by solving 𝑦 = 2𝑥 − 3 𝑓𝑜𝑟 𝑥). 2 𝑦+3 𝑦+3 Then 𝑓(𝑥 ) = 𝑓 ( ) =2[ ] − 3 = ( 𝑦 + 3) − 3 = 3 = 𝑦 2 2 Thus 𝒇 is onto and so is a one-to-one correspondence 12 Odd and Even functions The function 𝑦 = 𝑓 (𝑥 ) is even if 𝑓 (−𝑥) = 𝑓(𝑥 ) Similarly, the function 𝑦 = 𝑓 (𝑥 ) is odd if 𝑓 (−𝑥 ) = −𝑓(𝑥 ) Example Determine whether the following functions are even or odd functions. In each case find the zeros of the function. i. 𝑓( 𝑥 ) = 𝑥 3 − 𝑥 ii. 𝑔( 𝑥 ) = 𝑥 2 + 1 The function 𝑦 = 𝑓 (𝑥 ) is even if 𝑓 (−𝑥) = 𝑓(𝑥 ) and odd if 𝑓(−𝑥 ) = −𝑓(𝑥 ) i. 𝑓( 𝑥 ) = 𝑥 3 − 𝑥 Test: 𝑓(−𝑥 ) = (−𝑥)3 − (−𝑥 ) = −𝑥 3 + 𝑥 = −(𝑥 3 − 𝑥) This means the function 𝑓(𝑥 ) = 𝑥 3 − 𝑥 is odd Zeros 𝑥 3 − 𝑥 = 0, or 𝑥 (𝑥 2 − 1) = 𝑥 (𝑥 − 1)(𝑥 + 1) = 0 ⇒ 𝑥 = 0, 1, −1 ii. 𝑔( 𝑥 ) = 𝑥 2 + 1 Test: 𝑔(−𝑥 ) = (−𝑥)2 + 1 = 𝑥 2 + 1 = 𝑔(𝑥 ) This means the function 𝑔(𝑥 ) = 𝑥 2 + 1 is even Zeros: the function has no zeros since 𝑥 2 + 1 is positive for all 𝑥 13 Composite Functions If f is a function from X to Y and g is a function from Y to Z, then it is possible to combine them to obtain a function gf from X to Z. The function gf is called the composition of g and f. Definition: The composition of functions f and g is defined by (𝒇 𝒐 𝒈)(𝒙) = 𝒇(𝒈(𝒙))for all x in the domain of g such that 𝒈(𝒙) is in the domain of f. X Y Z f g a f(a) f(g(a)) b f(b) f(g(b)) c f(c) f(g(c)) x f(x) f(g(x)) (𝒇 𝒐 𝒈)(𝒙) Example If 𝑓(𝑥 ) = 𝑥 2 and 𝑔(𝑥 ) = 3𝑥 − 4 find i. (𝑓𝑜𝑔)(𝑥 ) and ii. (𝑔𝑜𝑓)(𝑥) iii. determine their domains. i. (𝑓 𝑜 𝑔)(𝑥 ) = 𝑓(𝑔(𝑥 )) = 𝑓(3𝑥 − 4) = (3𝑥 − 4)2 = 9(𝑥 )2 − 24𝑥 + 16 ii. (𝑔𝑜 𝑓)(𝑥 ) = 𝑔(𝑓(𝑥 )) = 𝑔(𝑥 2 ) = 3𝑥 2 − 4 Because g and f are both defined for all real numbers, so is 𝑓 𝑜 𝑔 and 𝑔 𝑜 𝑓 Note: The example demonstrates an important idea, namely, that the composition of functions is not a commutative operation. In other words, (𝑓 𝑜 𝑔)(𝑥) ≠ (𝑔𝑜 𝑓)(𝑥 ) for all functions f and g. however, there is a special class of functions for which (𝑓 𝑜 𝑔)(𝑥 ) = (𝑔𝑜 𝑓)(𝑥 ) 14 Inverse Functions If you have a function 𝑓, then you can try to define a new function 𝑓 −1 , the so-called inverse function of 𝑓, by the following prescription: f f (c ) −1 f c b f (b) a f (a ) a b c f ( a) f (b) f ( c) The graph of a function and its inverse are mirror images of each other For any given 𝑓 we say that 𝑦 = 𝑓 −1 (𝑥) if 𝑦 is the solution to the equation 𝑓(𝑦) = 𝑥. So to find 𝑦 = 𝑓 −1 (𝑥) you solve the equation 𝑥 = 𝑓(𝑥 ). If this is to define a function, then the prescribed rule must be unambiguous and the equation 𝑓(𝑦) = 𝑥.has to have a solution and cannot have more than one solution. Definition Let’s consider a one-to-one function f that assigns the value of f(x) in its range R to each x in its domain D. f g x f(x) x f(x) Domain Range Domain Range We can define a new function g that goes from R to D; it assigns f(x) in R back to x in D. The function f and g are called inverse functions of one another. 15 Note: Let f be a one-to-one function with a domain of X and a range of Y. A function g with a domain of Y and a range of X is called the inverse function of f if (𝒇 𝒐 𝒈)(𝒙) = 𝒙, for every x in Y and (𝒈 𝒐 𝒇)(𝒙) = 𝒙, for every x in X x+5 Example: Verify that f(x) = 4x − 5 and g(x) = , are inverse functions. 4 Because the set of real numbers is the domain and range of both functions, we know that the domain of f equals the range of g and the range of f equals the domain of g. Furthermore, 𝑥+5 𝑥+5 (𝑓 𝑜 𝑔)(𝑥 ) = 𝑓(𝑔(𝑥 )) = 𝑓 ( ) =4( )−5 = 𝑥 4 4 and (𝑔 𝑜 𝑓)(𝑥 ) = 𝑔(𝑓(𝑥 )) = 𝑔(4𝑥 − 5) = 4𝑥 −5+5 = 𝑥 4 Therefore, f and g are inverses of each other. Finding inverse functions The idea of inverse functions “undoing each other” provides the basis for a rather informa l approach to finding the inverse function. This informal approach may not work very well with more complex functions, but it does emphasize how inverse functions are related to each other. A more formal and systematic technique for finding the inverse of a function can be described as follows: 1. Replace the symbol 𝒇(𝒙)by y. 2. Interchange x and y 3. Solve the equation for y in terms of x 4. Replace y by 𝒇−𝟏 (𝒙) 16 Example Find the inverse of f(x) = x 2 − 2, where x ≥ 0 1. Replace the symbol 𝒇(𝒙)by y. 𝑦 = 𝑥 2 − 2, 𝑥 ≥ 0 2. Interchange x and y 𝑥 = 𝑦 2 − 2, 𝑦 ≥ 0 3. Solve the equation for y in terms of x 𝑥 = 𝑦 2 − 2 ⇒ 𝑦 = √ 𝑥 + 2, where 𝑥 ≥ −2 Replace y by 𝒇−𝟏 (𝒙) 𝑓 −1 (𝑥) = √ 𝑥 + 2, 𝑥 ≥ −2 17