Section Composite and Inverse Functions
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1 Math Section Page 1 Section Composite and Inverse Functions I. Composition of Functions A. If f and g are functions, then the composite function of f and g (written f g) is: (f g)( = f(g() The domain of f g is the set of all x in the domain of g such that g( is in the domain of f. B. With composition, we are, in effect, substituting a number into g(, finding out what y is, and then substituting that answer into f(. C. Examples - Let f( = 9 2x, g( = 5x + 2. find the following. 1. (f g)( First, we use the definition of composition to get: (f g)( = f(g() Now we will substitute into this equation what g( is equal to: (f g)( = f( 5x + 2) Next, we substitute 5x + 2 in for x in f(, EVEN THOUGH x IS REPEATED! (f g)( = 9 2( 5x + 2) Simplifying, we get: Answer: (f g)( = x 2. Now you try one: (g f)( Answer: (g f)( = x Note that composition, in general, is not commutative. 3. (f g)(3) Again, we start by using the definition of composition to get: (f g)(3) = f(g(3)) Substituting 3 for x in g(, we get: (f g)(3) = f( 5(3) + 2) = f( 13) We now substitute 13 in for x in f( to get: (f g)(3) = 9 2( 13) Simplifying, we get: Answer: (f g)(3) = Now you try one: (g f)(3) Answer: (g f)(3) = 13
2 Math Section Page 2 5. y = f( y = g( a. (f g)( 2) We start by using the definition of composition: (f g)( 2) = f(g( 2)) We now have to determine the value of y when x is 2 for the graph of g(: (f g)( 2) = f(2) We next look at the graph of f( and determine the value of y when x is 2: Answer: (f g)( 2) = 3 b. Now you try one: (g f)( 4) Answer: (g f)( 4) = 2 II. Inverse Properties A. Recall that for a real number A, the additive inverse was that real number B such that A + B = 0. B. For a real number A 0, the multiplicative inverse is that real number B such that AB = 1. C. For a function f(, the inverse function is that function g( such that ( f g) ( = x and ( g f ) ( = x. D. Verifying that functions are inverses of each other. f g (. If the answer is x, you are halfway there. 1. Do the composition ( ) 2. Now do the composition ( g f ) (. If this answer is also x, then f and g are inverse functions of each other. We then would write that g( = f -1 (. The "-1" is NOT an exponent. This notation means that we have the inverse function of f(. Note that f( is also g -1 (. E. Examples - Determine if f( and g( are inverses of each other. 1. f( = 3 x 4, g( = x f g (. We first do ( ) ( f g) ( = f(g() = f(x 3 + 4) Now substitute this in for x in f. 3 3 = 3 ( x 3 + 4) 4 = x = x 3 = x So this is half right. Now we do ( f ) ( g f ) ( = g(f() = g( 3 x 4 ) = ( 3 x 4 ) = x = x Answer: f( and g( are inverses. g (.
3 Math Section Page 3 2. Now you try one: f( = 5x 9, g( = Answer: f( and g( are not inverses. x III. Inverse Functions A. For a function f(, the inverse function is that function g( such that ( f g) ( = x and ( g f ) ( = x. B. Verifying that functions are inverses of each other. f g (. If the answer is x, you are halfway there. 1. Do the composition ( ) 2. Now do the composition ( g f ) (. If this answer is also x, then f and g are inverse functions of each other. We then would write that g( = f 1 (. The " 1" is NOT an exponent. This notation means that we have the inverse function of f(. Note that f( is also g 1 (. IV. Determining if a Function has an Inverse A. A function f( is one-to-one if for every y in the range there is only one x in the domain that corresponds to it. B. Horizontal Line Test: A function f( is not one-to-one if any horizontal line intersects the graph of f( in more than one point. C. If a function f( is one-to-one, then its inverse is also a function. When this occurs, we write the inverse function as f -1 ( (read "f inverse of x"). Note that this is the functional inverse, NOT the multiplicative inverse. D. Examples - Are these functions one-to-one? Answer: Not one-to-one. Answer: Yes one-to-one.
4 Math Section Page 4 3. Now you try one: Answer: Yes one-to-one. V. Finding the Inverse A. In general, to find the inverse of a relation, we switch x & y in the ordered pairs. Remember that x is the domain, y is the range. B. This means, geometrically, that the graph of a relation and its inverse are reflections of each other across the identity function line, f( = x. C. Finding the inverse of a function f( 1. Determine if the function is one-to-one. 2. Write y for f(. 3. Switch x & y. 4. Solve for y. 5. Write f 1 ( for y. 6. Verify by showing that ( f f 1) ( = ( f 1 f )( = x. 7. Remember: a. Domain of f is the range of f 1. b. Range of f is the domain of f 1. D. Examples - Find the inverse function. 1. f( = 4x 5 First, we write y for f(. y = 4x 5 Next, switch x & y. x = 4y 5 Now we solve for y. x + 5 x + 5 = 4y OR = y 4 Answer: f ( = x + Verify: ( f f 1) ( x ) = f(f 1 () = f 1 5 x + = x + 5 = x = x So this is ok. 1. You verify that ( f f )( = x Note that if we graph these, we make a table for the function that is the "easiest", then switch x & y to get the table for the inverse.
5 Math Section Page 5 2. f( = 3 x 5 First, we replace f( with y. y = 3 x 5 Now we switch x & y. x = 3 y 5 Now we solve for y. x 3 = y 5 OR x = y Answer: f 1 ( = x Verify: ( f 1 )( x ) = f f(f 1 () = f(x ) = ( 3 ) 1. You verify that ( f f )( = x 3. Now you try one: f( = x Answer: f 1 ( = 3 x 1 x = 3 3 x = x. So this is ok. VI. Graphing a function and its inverse. A. Remember that to find the inverse, we switch x & y. B. So if we make a table for f(, to get a table for f 1 (, we just switch x & y on the table. C. Examples - Graph f( and f 1 ( on the same set of axes. 1. f( = 4x 5, f 1 ( = 1 x + 5 Making a table for f( will be relatively easy, but f 1 ( doesn't look so nice! x f( = 4x Now graph both of these. Switch x & y on the table to get the table for f 1 (. f( x f 1 ( = 1 x f 1 (
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