Mathematics
Combining Functions
Combining functions is the process of taking two or more functions and creating a new function by applying one function to the output of another function. This is done by substituting the input of one function into the other function and simplifying the resulting expression. The resulting function is called a composite function.
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5 Key excerpts on "Combining Functions"
eBook - PDF- Linda Almgren Kime, Judith Clark, Beverly K. Michael(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
CHAPTER 9 New Functions from Old Overview Now we look at many ways of transforming or Combining Functions among all the families of functions we have studied. We start by transforming a function using stretching, com- pressing, shifting, or reflecting. The algebra of functions says that we can combine different functions through addition, subtraction, multiplication, or division. Certain power functions can be added to get polynomials, and polynomials can be divided to get rational functions. Composition of two functions is introduced and used to determine whether a function has an inverse. This final chapter closes with a collection of examples and Explore & Extends that weave together all the functions and techniques learned throughout your course. After reading this chapter, you should be able to • transform any function using stretches, compressions, shifting, and reflecting • combine any two functions using addition, subtraction, multiplication, and division • construct and interpret graphs of polynomials and their quotients, rational functions • compose two functions and look for inverse functions • use the families of functions and the algebraic tools introduced throughout the text to solve new problems 515 9.1 Transformations Finally, we look at the many ways we can create new functions from old, using examples from all the families of functions we have studied. This section considers transformations of func- tions and their graphs. What happens to the formula of a function when we shift, compress, or reflect the graph of the function? What happens to the graph of the function if we change the input or output of the function? Throughout the text, we have visualized the relationship between changes in the formulas of specific functions and changes in their graphs. Now we attach the language of functions to these changes and generalize to any function.
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.
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- 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 - PDFCollege Algebra
Building Concepts and Connections 2E
- Revathi Narasimhan(Author)
- 2019(Publication Date)
- XYZ Textbooks(Publisher)
For example, if profit is given in terms of dollars, another function must convert the profit function to a different currency. Successive evaluation of this series of two functions is known as a composition of functions and is of both practical and theoretical importance. Definition of a Composite Function The composition of functions f and g is a function that is denoted by f ∘ g and defined as ( f ∘ g )( x ) = f ( g ( x )) The domain of f ∘ g is the set of all x in the domain of g such that g ( x ) is in the domain of f . The function f ∘ g is called a composite function . Example 3 188 Chapter 2 Functions and Graphs Finding and Evaluating Composite Functions Let f ( s ) = s 2 + 1 and g ( s ) = − 2 s . a. Find an expression for ( f ∘ g )( s ) and give the domain of f ∘ g . b. Find an expression for ( g ∘ f )( s ) and give the domain of g ∘ f . c. Evaluate ( f ∘ g )( − 1). d. Evaluate ( g ∘ f )( − 1). Solution a. The composite function f ∘ g is defined as ( f ∘ g )( s ) = f ( g ( s )). Computing the quantity f ( g ( s )) is often the most confusing part. To make things easier, think of f ( s ) as f ( □ ), where the box can contain anything. Then proceed as follows: f ( □ ) = ( □ ) 2 + 1 Definition of f f ( g ( s ) ) = ( g ( s ) ) 2 + 1 Place g ( s ) in the box = ( − 2s ) 2 + 1 Substitute expression for g ( s ) = 4 s 2 + 1 Simplify: ( − 2 s ) 2 = 4 s 2 Thus, ( f ∘ g )( s ) = 4 s 2 + 1. Since the domain of g is all real numbers and the domain of f is also all real numbers, the domain of f ∘ g is all real numbers, ( −∞ , ∞ ). b. The composite function g ∘ f is defined as ( g ∘ f )( s ) = g ( f ( s )). Now, we compute g ( f ( s )). Thinking of g ( s ) as g ( □ ), we have g ( □ ) = − 2( □ ) Definition of g g ( f ( s ) ) = − 2( f ( s ) ) Place f ( s ) in the box = − 2( s 2 + 1 ) Substitute expression for f ( s ) = − 2 s 2 − 2 Simplify Thus, ( g ∘ f )( s ) = − 2 s 2 − 2. Note that ( f ∘ g )( s ) is not equal to ( g ∘ f )( s ).
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