Mathematics
Composition of Functions
Composition of functions is a mathematical operation that involves combining two or more functions to create a new function. The output of one function is used as the input for another function, and the resulting function is called the composite function. This operation is denoted by the symbol "∘".
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8 Key excerpts on "Composition of Functions"
eBook - PDF- Cynthia Y. Young(Author)
- 2018(Publication Date)
- Wiley(Publisher)
This is an example of a Composition of Functions, when the output of one function is the input of another function. It is commonly referred to as a function of a function. An algebraic example of this is the function y 5 "x 2 2 2 . Suppose we let g 1 x 2 5 x 2 2 2 and ƒ 1 x 2 5 !x . Recall that the independent variable in function notation is a placeholder. Since ƒ 1 u2 5 "1 u2 , then ƒ 1 g 1 x 22 5 "1 g 1 x 22 . Substituting the expression for g 1 x 2 , we find ƒ 1 g 1 x 22 5 "x 2 2 2 . The function y 5 "x 2 2 2 is said to be a composite function, y 5 ƒ 1 g 1 x 22 . Note that the domain of g 1 x 2 is the set of all real numbers, and the domain of ƒ 1 x 2 is the set of all nonnegative numbers. The domain of a composite function is the set of all x such that g 1 x 2 is in the domain of ƒ. For instance, in the composite function y 5 ƒ 1 g 1 x 22 , we know that the allowable inputs into ƒ are all numbers greater than or equal to zero. Therefore, we restrict the outputs of g 1 x 2 $ 0 and find the corresponding x-values. Those x-values are the only allowable inputs and constitute the domain of the composite function y 5 ƒ 1 g 1 x 22 . The symbol that represents Composition of Functions is a small open circle; thus 1 ƒ + g 21 x 2 5 ƒ 1 g 1 x 22 and is read aloud as “ƒ of g.” It is important not to confuse this with the multiplication sign: 1 ƒ ⋅ g 21 x 2 5 ƒ 1 x 2 g 1 x 2 . 1.4.2 SKILL Evaluate composite functions and determine the corresponding domains. 1.4.2 CONCEPTUAL Realize that the domain of a Composition of Functions excludes the values that are not in the domain of the inside function. ▼ C A U T I O N ƒ + g 2 ƒ ⋅ g 1.4 Combining Functions 153 154 CHAPTER 1 Functions and Their Graphs Composition of Functions Given two functions ƒ and g, there are two composite functions that can be formed.
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 ).
No longer available |Learn more- James Stewart, Lothar Redlin, Saleem Watson(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
SECTION 2.7 ■ Combining Functions 215 So far, we have used composition to build complicated functions from simpler ones. But in calculus it is useful to be able to “decompose” a complicated function into sim- pler ones, as shown in the following example. EXAMPLE 6 ■ Recognizing a Composition of Functions Given F1 x 2 ! 4 x 9 , find functions f and g such that F f g . SOLUTION Since the formula for F says to first add 9 and then take the fourth root, we let g 1 x 2 x 9 and f 1 x 2 ! 4 x Then 1 f g 21 x 2 f 1 g 1 x 22 Definition of f g f 1 x 9 2 Definition of g ! 4 x 9 Definition of f F1 x 2 Now Try Exercise 63 ■ ■ Applications of Composition When working with functions that model real-world situations, we name the variables us- ing letters that suggest the quantity being modeled. We may use t for time, d for distance, V for volume, and so on. For example, if air is being pumped into a balloon, then the radius R of the balloon is a function of the volume V of air pumped into the balloon, say, R f 1 V 2 . Also the volume V is a function of the time t that the pump has been working, say, V g 1 t 2 . It follows that the radius R is a function of the time t given by R f 1 g 1 t 22 . EXAMPLE 7 ■ An Application of Composition of Functions A ship is traveling at 20 km/h parallel to a straight shoreline. The ship is 5 km from shore. It passes a lighthouse at noon. (a) Express the distance s between the lighthouse and the ship as a function of d, the distance the ship has traveled since noon; that is, find f so that s f 1 d 2 . (b) Express d as a function of t, the time elapsed since noon; that is, find g so that d g 1 t 2 . (c) Find f g . What does this function represent? SOLUTION We first draw a diagram as in Figure 5. (a) We can relate the distances s and d by the Pythagorean Theorem.
eBook - PDF- Jerome Kaufmann, Karen Schwitters, , , Jerome Kaufmann, Karen Schwitters(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
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. Chapter 9 • Functions 470 Unless otherwise noted, all content on this page is © Cengage Learning. Definition 9.3 The Composition of Functions f and g is defined by ( f + g )( x ) 5 f ( g ( x )) for all x in the domain of g such that g ( x ) is in the domain of f . The left side, ( f + g )( x ) , of the equation in Defi-nition 9.3 can be read “the composition of f and g ,” and the right side, f ( g ( x )) , can be read “ f of g of x .” It may also be helpful for you to picture Definition 9.3 as two function machines hooked together to produce another function (often called a composite function ) as illustrated in Figure 9.34. Note that what comes out of the function g is substituted into the function f . Thus composition is sometimes called the substitution of functions. Figure 9.34 also vividly illustrates the fact that f + g is defined for all x in the domain of g such that g ( x ) is in the domain of f . In other words, what comes out of g must be capable of being fed into f . Let’s consider some examples. If f ( x ) 5 x 2 and g ( x ) 5 x 2 3 , find ( f + g )( x ) and determine its domain. Solution Applying Definition 9.3, we obtain ( f + g )( x ) 5 f ( g ( x )) 5 f ( x 2 3) 5 ( x 2 3) 2 Because g and f are both defined for all real numbers, so is f + g . If f ( x ) 5 ! x and g ( x ) 5 x 2 4 , find ( f + g )( x ) and determine its domain. Solution Applying Definition 9.3, we obtain ( f + g )( x ) 5 f ( g ( x )) 5 f ( x 2 4) 5 x 2 4 The domain of g is all real numbers, but the domain of f is only the nonnegative real num-bers. Thus g ( x ) , which is x 2 4 , has to be nonnegative.
eBook - PDF- Valentin Deaconu, Donald C. Pfaff(Authors)
- 2016(Publication Date)
- Chapman and Hall/CRC(Publisher)
62 A bridge to higher mathematics 4.5 Composition and inverse functions Definition 4.41. Suppose f and g are functions. We define a new function f ◦ g called the composition of f and g with domain {x ∈ dom(g) : g(x) ∈ dom(f )} such that (f ◦ g)(x) = f (g(x)). The domain of f ◦ g may be the empty set, in which case f ◦ g is the empty function (not so interesting). To avoid this situation, many times we assume that ran(g)=dom(f ). If f, g : X → X , then both functions f ◦ g and g ◦ f may be defined. In general, f ◦ g = g ◦ f . Example 4.42. Let f : [0, 4] → [0, 2], f (x) = √ x and let g : R → [−1, ∞), g(x) = x 2 − 1. Then f ◦ g : [− √ 5, −1] ∪ [1, √ 5] → [0, 2], (f ◦ g)(x) = x 2 − 1 and g ◦ f : [0, 4] → [−1, 3], (g ◦ f )(x) = x − 1. Example 4.43. The function h : (0, π/2) → R, h(x) = ln(tan x) is the compo- sition g ◦ f where g : (0, ∞) → R, g(y) = ln y and f : (0, π/2) → (0, ∞), f (x) = tan x. Notice that (f ◦ g)(z ) = tan(ln z ) makes sense only for z ∈ (1, e π/2 ) and f ◦ g = g ◦ f . Example 4.44. Let f : R → R, f (x) = x − 1 if x ≤ 1 x 2 if x > 1, g : R → R, g(x) = x/2 if x ≥ 2 x 3 if x < 2. Let us find f ◦ g and g ◦ f . Solution. We have (f ◦ g)(x) = g(x) − 1 if g(x) ≤ 1 (g(x)) 2 if g(x) > 1. Notice that g(x) ≤ 1 for x ≤ 1 and for x = 2; otherwise g(x) > 1. It follows that (f ◦ g)(x) = x 3 − 1 if x ≤ 1 x 6 if 1 < x < 2 0 if x = 2 x 2 4 if x > 2. On the other hand, (g ◦ f )(x) = f (x) 2 if f (x) ≥ 2 (f (x)) 3 if f (x) < 2. Functions 63 We have f (x) ≥ 2 for x ≥ √ 2 and f (x) < 2 for x < √ 2, hence (g ◦ f )(x) = x 2 2 if x ≥ √ 2 x 6 if 1 < x < √ 2 (x − 1) 3 if x ≤ 1. The operation of Composition of Functions preserves injectivity and surjec- tivity: Theorem 4.45. Let f : Y → Z and let g : X → Y . If f and g are one-to-one functions, then so is f ◦ g : X → Z . If f and g are onto, then f ◦ g : X → Z is also onto. Proof. Assume (f ◦ g)(x) = (f ◦ g)(x ′ ), so f (g(x)) = f (g(x ′ )) for x, x ′ ∈ X .
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Orange Apple(Publisher)
A function may also be described through its relationship to other functions, for example, as the inverse function or a solution of a differential equation. There are uncountably many different functions from the set of natural numbers to itself, most of which cannot be expressed with a formula or an algorithm. In a setting where they have numerical outputs, functions may be added and multiplied, yielding new functions. Collections of functions with certain properties, such as continuous functions and differentiable functions, usually required to be closed under certain operations, are called function spaces and are studied as objects in their own right, in such disciplines as real analysis and complex analysis. An important operation on functions, which distinguishes them from numbers, is the Composition of Functions. Overview Because functions are so widely used, many traditions have grown up around their use. The symbol for the input to a function is often called the independent variable or argument and is often represented by the letter x or, if the input is a particular time, by the letter t . The symbol for the output is called the dependent variable or value and is often represented by the letter y . The function itself is most often called f , and thus the notation y = f ( x ) indicates that a function named f has an input named x and an output named y . ________________________ WORLD TECHNOLOGIES ________________________ A function ƒ takes an input, x , and returns an output ƒ( x ). One metaphor describes the function as a machine or black box that converts the input into the output. The set of all permitted inputs to a given function is called the domain of the function. The set of all resulting outputs is called the image or range of the function. The image is often a subset of some larger set, called the codomain of a function.
Available until 7 Feb |Learn moreAbstract Algebra
An Inquiry Based Approach
- Jonathan K. Hodge, Steven Schlicker, Ted Sundstrom(Authors)
- 2013(Publication Date)
- Chapman and Hall/CRC(Publisher)
In this case, f ( x ) , the output of the function f , was used as the input for the function g . This idea motivates the formal definition of the composition of two functions. Definition A.11. Let A , B , and C be nonempty sets, and let f : A → B and g : B → C be functions. The composition of f and g is the function g ◦ f : A → C defined by ( g ◦ f )( x ) = g ( f ( x )) for all x ∈ A . We often refer to the function g ◦ f as a composite function . Activity A.12. Let A = { 1 , 2 , 3 } , B = { a,b,c,d } , and C = { s,t } . Define f : A → B by f (1) = a,f (2) = b,f (3) = c, g : A → B by g (1) = c,g (2) = d,g (3) = c, and h : B → C by h ( a ) = s,h ( b ) = s,h ( c ) = t,h ( d ) = s. (a) Find the images of the elements in A under the function f ◦ h . (b) Find the images of the elements in A under the function g ◦ h . (c) Is f ◦ h an injection? Is f ◦ h a surjection? Explain. (d) Is g ◦ h an injection? Is g ◦ h a surjection? Explain. In Activity A.12, we asked questions about whether certain composite functions were injections and/or surjections. In mathematics, it is typical to explore whether certain properties of an object transfer to related objects. In particular, we might want to know whether or not the composite of two injective functions is also an injection. (Of course, we could ask a similar question for surjections.) These types of questions are explored in the next activity. Activity A.13. Let the sets A , B , C , and D be as follows: A = { a,b,c } , B = { p,q,r } , C = { u,v,w,x } , and D = { u,v } . (a) Construct a function f : A → B that is an injection and a function g : B → C that is an injection. In this case, is the composite function g ◦ f : A → C an injection? Explain. (b) Construct a function f : A → B that is a surjection and a function g : B → D that is a surjection. In this case, is the composite function g ◦ f : A → D a surjection? Explain. (c) Construct a function f : A → B that is a bijection and a function g : B → A that is a bijection.
eBook - PDFFunctions Modeling Change
A Preparation for Calculus
- Eric Connally, Deborah Hughes-Hallett, Andrew M. Gleason(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
. . 367 Summary for Section 10.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 10.4 Combinations of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 The Difference of Two Functions Defined by Formulas: A Measure of Prosperity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 The Sum and Difference of Two Functions Defined by Graphs . . . . . . . . . . . . . . . . . . . . 372 Factoring a Function’s Formula into a Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 The Quotient of Functions Defined by Formulas and Graphs: Prosperity . . . . 374 The Quotient of Functions Defined by Tables: Per-Capita Crime Rate . . . . . . . 375 Summary for Section 10.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 STRENGTHEN YOUR UNDERSTANDING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 Chapter 10 COMPOSITIONS, INVERSES, AND COMBINATIONS OF FUNCTIONS 350 Chapter 10 COMPOSITIONS, INVERSES, AND COMBINATIONS OF FUNCTIONS 10.1 REVISITING Composition of Functions In Section 2.5, we introduced the composition of two functions and as follows: The function (()), called the composition of with , is defined by using the output of the function as the input to . The composite function (()) is defined only for values of in the domain of whose () values are in the domain of . How Does Composition Arise? A drug has the side effect of raising a patient’s heart rate. If the amount of drug in the patient’s body decreases with time, then the heart rate decreases also. The heart rate is given as a function of time by a Composition of Functions.
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