SECTION 5.1 Composite Functions 273 Figure 1 DEFINITION Composite Function Given two functions f and g , the composite function , denoted by f g (read as “ f composed with g ”), is defined by f g x f g x ( ) ( ) ( ) ( ) = The domain of f g is the set of all numbers x in the domain of g for which g x( ) is in the domain of f. Figure 2 Domain of f ° g Range of f °g Range of g Domain of f Domain of g f ° g g f g x x g(x) g(x) f(g(x)) Range of f Luis P. Villarreal Luis Perez Villarreal began his college studies at a local community college and eventually studied biological chemistry in graduate school. He is now a research scientist at University of California, with a particular interest in how you ‘program’ biological things. He is also interested in how you can write programs to re-engineer what cells do in an organism, like gene therapy. 5.1 Composite Functions Now Work the ‘Are You Prepared?’ problems on page 277. • Find the Value of a Function (Section 2.1, pp. 65–67) • Domain of a Function (Section 2.1, pp. 69–73) PREPARING FOR THIS SECTION Before getting started, review the following: OBJECTIVES 1 Form a Composite Function (p. 273) 2 Find the Domain of a Composite Function (p. 274) 1 Form a Composite Function Suppose that an oil tanker is leaking oil and you want to determine the area of the circular oil patch around the ship. See Figure 1. It is determined that the radius of the circular patch of oil around the ship is increasing at a rate of 3 feet per minute. Then the radius r of the oil patch at any time t , in minutes, is given by r t t3 . ( ) = So after 20 minutes, the radius of the oil patch is r 20 3 20 60 feet. ( ) = ⋅ = The area A of a circle is a function of the radius r given by A r r .2 π ( ) = The area of the circular patch of oil after 20 minutes is A 60 60 3600 2 π π ( ) ( ) = = square feet. Note that r 60 20 , ( ) = so A A r 60 20 . ( ) ( ( )) = The argument of the function A is the output of the function r . In general, the area of the oil patch can be expressed as a function of time t by evaluating A r t ( ( )) and obtaining A r t A t t t 3 3 9 . 2 2 π π ( ( )) ( ) ( ) = = = The function A r t ( ( )) is a special type of function called a composite function. As another example, consider the function y x2 3 . 2 ( ) = + Let y f u u2 ( ) = = and u g x x2 3. ( ) = = + Then by a substitution process, the original function is obtained as follows: y f u f g x x2 3 . 2 ( ) ( ) ( ) ( ) = = = + In general, suppose that f and g are two functions and that x is a number in the domain of g . Evaluating g at x yields g x( ) . If g x( ) is in the domain of f, then evaluating f at g x( ) yields the expression f g x ( ) ( ) . The correspondence from x to f g x ( ) ( ) is called a composite function f g. Look carefully at Figure 2. Only those values of x in the domain of g for which g x( ) is in the domain of f can be in the domain of f g. The reason is if g x( ) is not in the domain of f, then f g x ( ) ( ) is not defined. Because of this, the domain of f g is a subset of the domain of g ; the range of f g is a subset of the range of f.
RkJQdWJsaXNoZXIy NjM5ODQ=