Mathematics
Composite Functions
Composite functions are functions that are formed by combining two or more functions. The output of one function becomes the input of another function. The composition of functions is denoted by the symbol "∘".
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8 Key excerpts on "Composite 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 - PDF- Ron Larson(Author)
- 2021(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. 1.8 Combinations of Functions: Composite Functions 79 GO DIGITAL In Examples 5 and 6, you formed the composition of two given functions. In calculus, it is also important to be able to identify two functions that make up a given composite function. For example, the function h(x) = (3x - 5) 3 is the composition of f (x) = x 3 and g(x) = 3x - 5. That is, h(x) = (3x - 5) 3 = [g(x)] 3 = f (g(x)). Basically, to “decompose” a composite function, look for an “inner” function and an “outer” function. In the function h above, g(x) = 3x - 5 is the inner function and f (x) = x 3 is the outer function. EXAMPLE 7 Writing a Composite Function Write the function h(x) = 1 (x - 2) 2 as a composition of two functions. Solution One way to write h as a composition of two functions is to let g(x) = x - 2 be the inner function and let f (x) = 1 x 2 = x -2 1 a n = a -n be the outer function. Then you can write h as h(x) = 1 (x - 2) 2 = (x - 2) -2 = f (x - 2) = f (g(x)). Checkpoint Audio-video solution in English & Spanish at LarsonPrecalculus.com Write the function h(x) = 3 √8 - x 5 as a composition of two functions. EXAMPLE 6 Finding the Domain of a Composite Function Find the domain of f ∘ g for the functions f (x) = x 2 - 9 and g(x) = √9 - x 2 . Graphical Solution 4 - 10 - 4 2 The x-coordinates of points on the graph extend from - 3 to 3. So, the domain of f ° g is [- 3, 3]. Algebraic Solution Find the composition of f with g. ( f ∘ g)(x) = f ( g(x)) Definition of f ∘ g = f ( √9 - x 2 ) Definition of g(x) = ( √9 - x 2 ) 2 - 9 Definition of f (x) = 9 - x 2 - 9 ( n √a ) n = a = -x 2 Simplify.
eBook - PDF- Paul A. Calter, Michael A. Calter(Authors)
- 2011(Publication Date)
- Wiley(Publisher)
Rearranging gives us ◆◆◆ Composite Functions Just as we can substitute a constant or a variable into a given function, so we can substitute a function into a function. ◆◆◆ Example 36: If find (a) (b) (c) Solution: (a) (b) (c) ◆◆◆ Being made up of the two functions and the function , which we read “g of f of x,” is called a composite function. If we think of a function as a machine, it is as if we are using the output of the function machine f as the input of a second function machine g. We thus obtain by replacing x in by the function f(x). g(x) g[f(x)] S g[f(x)] g S f(x) S f x S f(x) g[f(x)] f(x), g(x) g[f(x)] f(x) 1 g(z 2 ) z 2 1 g(2) 2 1 3 g[f(x)] g(z 2 ) g(2) g(x) x 1, 3x 2 2x y 0 f(x, y) 0. y 3x 2 2x y 9 x 3 3y 27 x y f(x). x 27 3y x y 3 2 2x y 3 x f(y). y 2x 3 148 Chapter 4 ◆ Functions ◆◆◆ Example 37: Given the functions and write the com- posite function Solution: In the function we replace x by since ◆◆◆ As we have said, the notation means to substitute into the function On the other hand, the notation means to substitute into In general, will not be the same as ◆◆◆ Example 38: Given and find the following: (a) (b) (c) (d) Solution: (a) (b) (c) (d) Notice that here is not equal to ◆◆◆ Inverse of a Function Consider a function f that, given a value of x, returns some value of y. If y is now put into a function g that reverses the operations performed in f so that its output is the original x, then g is called the inverse of f. The inverse of a function is often designated by Do not confuse with ◆◆◆ Example 39: Two such inverse operations are “cube” and “cube root.” ◆◆◆ Thus if a function has an inverse that reverses the operations in then the composite of and should have no overall effect.
No longer available |Learn more- James Stewart, Lothar Redlin, Saleem Watson(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
For instance, the composite function f g h is found by first applying h, then g, and then f as follows: 1 f g h 21 x 2 f 1 g 1 h1 x 222 EXAMPLE 5 ■ A Composition of Three Functions Find f g h if f 1 x 2 x/ 1 x 1 2 , g1 x 2 x 10 , and h1 x 2 x 3. SOLUTION 1 f g h 21 x 2 f 1 g 1 h1 x 222 Definition of f g h f 1 g 1 x 3 22 Definition of h f 11 x 3 2 10 2 Definition of g 1 x 3 2 10 1 x 3 2 10 1 Definition of f Now Try Exercise 59 ■ The graphs of f and g of Example 4, as well as those of f g, g f, f f, and g g, are shown below. These graphs indicate that the operation of composi- tion can produce functions that are quite different from the original functions. f$g g$f f$f g$g f g Copyright 2017 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. 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 .
eBook - PDF- James Stewart, Lothar Redlin, Saleem Watson, , James Stewart, Lothar Redlin, Saleem Watson(Authors)
- 2015(Publication Date)
- Cengage Learning EMEA(Publisher)
SECTION 2.7 ■ Combining Functions 251 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 F 1 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 F 1 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 mi/h parallel to a straight shoreline. The ship is 5 mi 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- R. Gustafson, Jeff Hughes(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
Such chains of dependence are analyzed mathematically as composi- tion of functions. Suppose that y 5 f sxd and y 5 gsxd define two functions. Any number x in the domain of g will produce a corresponding value gsxd in the range of g. If gsxd is in the domain of function f, then gsxd can be substituted into f, and a correspond‑ ing value f sgsxdd will be determined. This two‑step process defines a new function, called a composite function, denoted by f + g. (See Figure 3‑52.) Domain of g Range of f Range of g Domain of f f ° g g f x g(x) f (g(x)) FIGURE 3-52 Self Check 5 Composite Function The composite function f + g is defined by s f + gdsxd 5 f s gsxdd The domain of f + g consists of all those numbers in the domain of g for which gsxd is in the domain of f. To illustrate the previous definition, we consider the functions f sxd 5 5x 1 1 and gsxd 5 4x 2 3 and find s f + gdsxd and sg + f dsxd. s f + gdsxd 5 f s gsxdd sg + f dsxd 5 gs f sxdd 5 f s4x 2 3d 5 gs5x 1 1d 5 5s4x 2 3d 1 1 5 4s5x 1 1d 2 3 5 20x 2 14 5 20x 1 1 Since we get different results, the composition of functions is not commutative. We have seen that a function can be represented by a machine. If we put a num‑ ber from the domain into the machine (the input), a number from the range comes out (the output). For example, if we put 2 into the machine shown in Figure 3‑53(a), Take Note Note that for this example s f + gdsxd Þ sg + f dsxd. Copyright 2017 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. 353 Section 3.4 Operations on Functions the number f s2d 5 5s2d 2 2 5 8 comes out.
eBook - PDF- Peter Grossman(Author)
- 2017(Publication Date)
- Red Globe Press(Publisher)
If we think of f and g as machines with input and output, we could imagine linking them together so that the output of f becomes the input of g : This will work only if the output from f belongs to the domain of g . In order to ensure that this is always the case, we will assume that the 103 6 Functions Figure 6.2 codomain of f equals the domain of g . Specifically, let A , B and C be arbitrary sets, and let f A B : ® and g B C : ® . We can now think of the combination of the two machines as a single machine with input x A Î and output g f x C ( ( )) Î . This new machine corresponds to a function from A to C , called the composite function of f and g . The formal definition follows. Definition Let f A B : ® and g B C : ® be functions. The composite function of f and g is the function: g f A C g f x g f x o o : ,( )( ) ( ( )) ® = Notice that g f o needs to be read from right to left: it means first apply f , then apply g to the result. Although real-valued functions are only of minor importance for our purposes, we will use them in our first example because you are likely to be more familiar with them. Example 6.2.1 Let f : R R ® , f ( x ) = x 2 and g : R R ® , g ( x ) = 3 x – 1. Find f g o and g f o . Solution Note firstly that the composite function f g o exists because the codomain of g equals the domain of f . Similarly, g f o exists because the codomain of f equals the domain of g . The function f g o is found as follows: f g f g x f g x f x x o o : ,( )( ) ( ( )) ( ) ( ) R R ® = = -= -3 1 3 1 2 where the last line is obtained by substituting 3 x – 1 in place of x in the formula for f ( x ). The function g f o is obtained in a similar manner: g f g f x g f x g x x o o : ,( )( ) ( ( )) ( ) R R ® = = = -2 2 3 1 The two Composite Functions in the last example can be depicted in the following way: 104 Discrete mathematics for computing Example 6.2.2 Let X be the set of all finite non-empty strings of characters.
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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