Mathematics
Inverse functions
Inverse functions are pairs of functions that "undo" each other. If a function f(x) maps an input x to an output y, its inverse function, denoted as f^-1(y), maps y back to x. In other words, the inverse function reverses the action of the original function, allowing us to retrieve the original input from the output.
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10 Key excerpts on "Inverse functions"
eBook - PDFA Course of Mathematical Analysis
International Series of Monographs on Pure and Applied Mathematics
- A. F. Bermant, I. N. Sneddon, S. Ulam, M. Stark(Authors)
- 2016(Publication Date)
- Pergamon(Publisher)
5. Inverse functions. Power, Exponential and Logarithmic Functions 20. The concept of inverse function. Let y be given as a function of x: y = f(x). (A) We can regard y as the independent variable in this functional relationship; x then becomes a function of y : * =
eBook - PDF- Mary Jane Sterling(Author)
- 2023(Publication Date)
- For Dummies(Publisher)
4 Equations and Applications IN THIS PART . . . Become acquainted with inverse trig functions. Identify the domains and ranges of the inverse trig functions. Recognize the pairings of the quadrants used by each inverse function. Solve trig equations using identities and Inverse functions. Write expressions to include infinitely many answers. Find the areas of triangles using trig functions in the formulas. CHAPTER 14 Investigating Inverse Trig Functions 233 Chapter 14 Investigating Inverse Trig Functions A s thrilling and fulfilling as the original six trig functions are, they just aren’t complete without their inverses. An inverse trig function behaves like the inverse of any other type of function — it undoes what the original function did. In mathematics, functions can have inverses if they’re one-to-one, meaning each output value occurs only once. This whole inverse idea is going to take some fast talking when it comes to trig functions, because they keep repeat- ing values over and over as angles are formed with every full rotation of the circle — so you’re going to wonder how these functions and inverses can be one- to-one. If you need a refresher on basic Inverse functions, just refer to the section on inverses in the Appendix for the lowdown on them and how you determine one. Writing It Right You use inverse trig functions when you want to know what angle is involved in equations such as sin x 1 2 or sec 2 x , or tan 2 1 x . In typical algebra equations, you can solve for the value of x by dividing each side of the equation by the coef- ficient or by adding the same thing to each side, and so on. But you can’t do that with the function sin x 1 2 . IN THIS CHAPTER » Acquainting yourself with inverse notation » Setting limits on inverse trig functions » Determining domain and range of inverse trig functions
eBook - PDF- Ron Larson, Bruce Edwards(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
CONCEPT CHECK 1. Reflective Property of Inverse functions Describe the relationship between the graph of a function and the graph of its inverse function. 2. Domain of an Inverse Function The function f has an inverse function f -1 . Is the domain of f the same as the domain of f -1 ? Explain. 3. Inverse Trigonometric Function Describe the meaning of arccos x in your own words. 4. Restricted Domain What is a restricted domain? Why are restricted domains necessary to define inverse trigonometric functions? Matching In Exercises 5–8, match the graph of the function with the graph of its inverse function. [The graphs of the Inverse functions are labeled (a), (b), (c), and (d).] (a) 1 2 3 4 5 1 2 3 - 2 - 1 - 3 x y (b) 2 4 4 6 6 8 - 4 x - 2 - 4 y (c) x 2 3 4 2 1 - 1 - 2 - 2 - 4 y (d) 1 2 3 1 2 3 - 2 - 3 - 3 x - 2 y 5. 1 2 2 3 4 - 1 - 2 - 2 - 4 x y 6. 4 2 4 6 6 8 8 - 4 x - 2 - 4 y 7. 1 2 3 1 2 3 - 2 - 1 - 3 - 3 x - 2 y 8. 1 2 3 1 2 3 - 2 - 3 x y Verifying Inverse functions In Exercises 9–16, show that f and g are Inverse functions (a) analytically and (b) graphically. 9. f (x) = 5x + 1, g(x) = x - 1 5 10. f (x) = 3 - 4x, g(x) = 3 - x 4 11. f (x) = x 3 , g(x) = 3 radical.alt2x 12. f (x) = 3 radical.alt2x - 3 , g(x) = 3 + x 3 13. f (x) = radical.alt2x - 4 , g(x) = x 2 + 4, x ≥ 0 14. f (x) = 16 - x 2 , x ≥ 0, g(x) = radical.alt216 - x 15. f (x) = 1 x , g(x) = 1 x 16. f (x) = 1 1 + x , x ≥ 0, g(x) = 1 - x x , 0 < x ≤ 1 Using the Horizontal Line Test In Exercises 17 and 18, use the Horizontal Line Test to determine whether the function is one-to-one on its entire domain and therefore has an inverse function. To print an enlarged copy of the graph, go to MathGraphs.com. 17. f (θ ) = sin θ 18. f (x) = 5x - 3 1 π 2 π 2 3 θ y - 2 - 1 - 3 - 1 - 2 1 1 2 x y The Existence of an Inverse Function In Exercises 19–24, determine whether the function is one-to-one on its entire domain and therefore has an inverse function. 19. f (x) = 2 - x - x 3 20. f (x) = x 4 4 - 2x 2 21.
eBook - PDF- Ron Larson(Author)
- 2021(Publication Date)
- Cengage Learning EMEA(Publisher)
In this text, whenever f -1 is written, it always refers to the inverse function of the function f and not to the reciprocal of f (x). If the function g is the inverse function of the function f, then it must also be true that the function f is the inverse function of the function g. So, it is correct to say that the functions f and g are Inverse functions of each other. EXAMPLE 2 Verifying Inverse functions Which of the functions is the inverse function of f (x) = 5 x - 2 ? g(x) = x - 2 5 h(x) = 5 x + 2 Solution By forming the composition of f with g, you have f (g(x)) = f ( x - 2 5 ) = 5 ( x - 2 5 ) - 2 = 25 x - 12 ≠ x. This composition is not equal to the identity function x, so g is not the inverse function of f. By forming the composition of f with h, you have f (h(x)) = f ( 5 x + 2 ) = 5 ( 5 x + 2 ) - 2 = 5 ( 5 x ) = x. So, it appears that h is the inverse function of f. Confirm this by showing that the composition of h with f is also equal to the identity function. h( f (x)) = h ( 5 x - 2 ) = 5 ( 5 x - 2 ) + 2 = x - 2 + 2 = x Check to see that the domain of f is the same as the range of h and vice versa. Checkpoint Audio-video solution in English & Spanish at LarsonPrecalculus.com Which of the functions is the inverse function of f (x) = x - 4 7 ? g(x) = 7x + 4 h(x) = 7 x - 4 Definition of Inverse Function Let f and g be two functions such that f (g(x)) = x for every x in the domain of g and g( f (x)) = x for every x in the domain of f. Under these conditions, the function g is the inverse function of the function f. The function g is denoted by f -1 (read “ f -inverse”). So, f ( f -1 (x)) = x and f -1 ( f (x)) = x. The domain of f must be equal to the range of f -1 , and the range of f must be equal to the domain of f -1 . Copyright 2022 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
eBook - PDF- Ken Binmore, Joan Davies(Authors)
- 2002(Publication Date)
- Cambridge University Press(Publisher)
Seven Inverse functions We begin this chapter with a fairly careful account of the inverse of a function of one variable. We have already studied the inverses of basic elementary functions in Chapter 2 . We now develop the idea of a ‘local inverse’ of a function of one variable, to address the situation when more than one value of the independent variable x is mapped by the function to the same value of the dependent variable y . This follows on from the power functions with their inverses, the root functions and we define the ‘inverse trigonometric functions’. The chapter continues with local Inverse functions in the vector case and applies this work to general coordinate systems. The ideas are explained at some length but it would be inappropriate to spend a lot of time on these sections if difficulties are being encountered elsewhere. For most purposes it is enough to have some understanding of why things go wrong when the Jacobian is zero and to be able to change the variables in differential operators as described in § 7.5 . 7.1 Local inverses of scalar valued functions We say that f − 1 : J → I is a local inverse for f : R → R if it is true that x = f − 1 ( y ) if and only if y = f ( x ) provided that x ∈ I and y ∈ J . The existence of the local inverse therefore requires that, for each y ∈ J , the equation y = f ( x ) has a unique solution x ∈ I . y J ξ y = f ( x ) y y y = f ( x ) x = f –1 ( y ) x x I η If ξ is an interior point of the interval I and η = f (ξ) is an interior point of the interval J , we say that f − 1 : J → I is a local inverse for f at the point x = ξ . For obvious reasons, we denote a local inverse function at the point x = ξ by f − 1 , but it is important to realise that there are many functions that can play this role. 235 Chapter 7. Inverse functions Example 1 Although the function f : R → R defined by y = f ( x ) = x 2 has no inverse function, it does have a local inverse function at any x = 0.
eBook - PDFThe Calculus Lifesaver
All the Tools You Need to Excel at Calculus
- Adrian Banner(Author)
- 2009(Publication Date)
- Princeton University Press(Publisher)
C h a p te r 10 Inverse functions and Inverse Trig Functions In the previous chapter, we looked at exponentials and logarithms. We got a lot of mileage out of the fact that e x and ln( x ) are inverses of each other. In this chapter, we’ll look at some more general properties of Inverse functions, then examine inverse trig functions (and their hyperbolic cousins) in greater detail. Here’s the game plan: • using the derivative to show that a function has an inverse; • finding the derivative of Inverse functions; • inverse trig functions, one by one; and • inverse hyperbolic functions. 10.1 The Derivative and Inverse functions In Section 1.2 of Chapter 1, we reviewed the basics of Inverse functions. I strongly suggest you take a quick look over that section before reading further, familiarizing yourself with the general idea. Now that we know some calculus, we can say more. In particular, we’re going to explore two connections between derivatives and Inverse functions. 10.1.1 Using the derivative to show that an inverse exists Suppose that you have a differentiable function f whose derivative is always positive. What do you think the graph of this function looks like? Well, the slope of the tangent has to be positive everywhere, so the function can’t dip up and down: it has to go upward as we look from left to right. In other words, the function must be increasing . We’ll prove this fact in the next chapter (see Section 11.3.1 and also Sec-tion 11.2), but it at least seems clear that it should be true. In any case, if our function f is always increasing, then it must satisfy the horizontal line test. No horizontal line could possibly hit the graph of y = f ( x ) twice. Since the horizontal line test is satisfied by f , we know that f has an inverse. This has given us a nice strategy for showing that a function has an inverse: show that its derivative is always positive on its domain.
eBook - PDF- Valentin Deaconu, Donald C. Pfaff(Authors)
- 2016(Publication Date)
- Chapman and Hall/CRC(Publisher)
You already met real functions of real variables in calculus, like f (x) = x 2 , g(x) = ln x or h(x) = tan x. The domain and the set of values for these functions are subsets of R, and a function is defined as a formula (or algorithm) which associates to each input in the domain a precise output. The domain of f is R and the set of values is [0, ∞). The domain of g is (0, ∞) and the set of values is R. The domain of h is R \ {(2k + 1) π 2 : k ∈ Z} and the set of values is R. We will need to work with more general functions among all kinds of sets, not just subsets of the real numbers. Even though a function is a particular case of a relation, we study functions first and define relations in the next chapter. After giving the precise defini- tion of a function using its graph, we introduce operations and give several examples of functions. A given function determines two new functions, called the direct image and the inverse image, where the inputs and the outputs are sets. Sometimes it is necessary to shrink or enlarge the domain of a function, giving rise to restrictions and extensions. We also discuss one-to-one and onto functions, composition, and Inverse functions. We conclude with families of sets and the axiom of choice, necessary in many proofs. 4.1 Definition and examples of functions Here is the formal definition of a function. Definition 4.1. A function from a set X to a set Y is a subset f of the Cartesian product X × Y such that for all x ∈ X there is a unique y ∈ Y with 〈x, y〉 ∈ f . The set X is called the domain of f , denoted dom(f ), and the set {y ∈ Y : ∃ x ∈ X with 〈x, y〉 ∈ f } 51 52 A bridge to higher mathematics is called the range of f , denoted ran(f ). The set Y is the set where f takes values, also called the codomain of f . Note that the range ran(f ) may be a proper subset of the codomain Y . We write f : X → Y , and for each x ∈ X the unique element y ∈ Y such that 〈x, y〉 ∈ f is denoted f (x).
eBook - PDFDiscrete Mathematics
Mathematical Reasoning and Proof with Puzzles, Patterns, and Games
- Douglas E. Ensley, J. Winston Crawley(Authors)
- 2011(Publication Date)
- Wiley(Publisher)
27. In your own words, explain how you can tell from a list of ordered pairs whether the relation with that list as its rule is a function. 28. Explain how you can tell from a list of ordered pairs whether the inverse of the relation with that list as its rule is a function. 4.2 The Composition Operation The idea of composition of functions and relations is a very basic one. Mathematics is largely about how complex concepts, structures, and properties can be built logically out of simpler ones. Since functions and relations are fundamental structures in mathematics, it stands to reason that combinations of two or more of these structures could be important. Composition of Functions Before being swept away by a formal definition, let us consider an example of how composition naturally arises in the English language. One rule that relates pairs of people is the “husband of” relation. Another rule of this type is the “mother of” relation. These relations can be combined to give two distinct meanings: • The relation “mother of the husband of” associates a woman with her mother-in- law. • The relation “husband of the mother of” associates any person with his or her father or stepfather. The preposition “of” naturally ties together English clauses in the same way that composition ties together mathematical functions, as we see in our formal definition. Definition If f : A → B and g : B → C , then we can build a new function called (g ◦ f ) that has domain A and codomain C , and that follows the rule (g ◦ f )(x) = g( f (x)). We call (g ◦ f ), read “g of f ,” the composition of g with f . 4.2 The Composition Operation 269 The double of the square root g f Description order for (g ° f ) g ( f ( x )) Second First Evaluation order for (g ° f ) Figure 4-22 How to read (g ◦ f ). Example 1 Given the function f : R ≥0 → R defined by the rule f (x) = √ x, and the function g : R → R defined by the rule g( y) = 2 · y, describe the domain, codomain, and rule for the function (g ◦ f ).
eBook - PDF- Cynthia Y. Young(Author)
- 2018(Publication Date)
- Wiley(Publisher)
The symmetry about the line y 5 x tells us that the roles of x and y interchange. Therefore, if we start with every point 1 x, y 2 that lies on the graph of a function, then every point 1 y, x 2 lies on the graph of its inverse. Algebraically, this corresponds to interchanging x and y. Finding the inverse of a finite set of ordered pairs is easy: Simply interchange the x- and y-coordinates. Earlier, we found that if h 1 x 2 5 51 21, 0 2 , 1 1, 2 2 , 1 3, 4 26 , then h 21 1 x 2 5 51 0, 21 2 , 1 2, 1 2 , 1 4, 3 26 . But how do we find the inverse of a function defined by an equation? Recall the mapping relationship if ƒ is a one-to-one function. This relationship implies that ƒ 1 x 2 5 y and ƒ 21 1 y 2 5 x. Let’s use these two identities to find the inverse. Now consider the function defined by ƒ 1 x 2 5 3x 2 1. To find ƒ 21 , we let ƒ 1 x 2 5 y, which yields y 5 3x 2 1. Solve for the variable x: x 5 1 3 y 1 1 3 . Recall that ƒ 21 1 y 2 5 x, so we have found the inverse to be ƒ 21 1 y 2 5 1 3 y 1 1 3 . It is customary to write the independent variable as x, so we write the inverse as ƒ 21 1 x 2 5 1 3 x 1 1 3 . Now that we have found the inverse, let’s confirm that the properties ƒ 21 1 ƒ 1 x 22 5 x and ƒ A ƒ 21 1 x 2B 5 x hold. ƒ A ƒ 21 1 x 2B 5 3 a 1 3 x 1 1 3 b 2 1 5 x 1 1 2 1 5 x ƒ 21 1 ƒ 1 x 22 5 1 3 1 3x 2 1 2 1 1 3 5 x 2 1 3 1 1 3 5 x 1.5.4 SKILL Find the inverse of a function. 1.5.4 CONCEPTUAL Understand why a function must be one-to-one in order for its inverse to exist. x ƒ ƒ 1 ƒ(x) Domain of ƒ Range of ƒ ƒ -1 (y) y Range of ƒ 1 Domain of ƒ 1 YOUR TURN Given the graph of a function ƒ, plot the inverse function. x y –5 5 –5 5 ▼ ▼ A N S W E R x y –5 5 –5 5 FINDING THE INVERSE OF A FUNCTION Let ƒ be a one-to-one function. Then the following procedure can be used to find the inverse function ƒ 21 if the inverse exists. STEP PROCEDURE EXAMPLE 1 Let y 5 ƒ1 x2 . ƒ1 x2 5 23x 1 5 y 5 23x 1 5 2 Solve the resulting equation for x in terms of y (if possible).
eBook - PDF- Douglas Smith, Maurice Eggen, Richard St. Andre, , Douglas Smith, Maurice Eggen, Richard St. Andre, , Douglas Smith, Maurice Eggen, Richard St. Andre(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
The next result gives a simple, practical method using composition to deter-mine whether a given function is the inverse of a function. Copyright 2015 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 234 CHAPTER 4 Functions Theorem 4.4.4 If f is a function from A to B and g is a function from B to A , then (a) g = f − 1 if and only if g ◦ f = I A and f ◦ g = I B . (b) If f is a one-to-one correspondence, then g = f − 1 if and only if g ◦ f = I A or f ◦ g = I B . Proof. (a) If g = f − 1 , then g ◦ f = I A and f ◦ g = I B , by Theorem 4.2.3. We use the fact that Rng ( f ) = Dom ( f − 1 ) = B . Assume now that g ◦ f = I A and f ◦ g = I B . Then f is one-to-one by Theorem 4.3.4, and f maps onto B by Theorem 4.3.2. Thus, f − 1 is a function on B and f − 1 = f − 1 ◦ I B = f − 1 ◦ ( f ◦ g ) = ( f − 1 ◦ f ) ◦ g = I A ◦ g = g . (b) See Exercise 5. ◾ Theorem 4.4.4(a) captures the essential idea of an inverse—whatever f does to a domain element x , applying the inverse to f ( x ) takes you right back to x .
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