d i E Inverse Functions a l l u d Dr. Abdulla Eid b A - PowerPoint PPT Presentation

Section 2.4 d i E Inverse Functions a l l u d Dr. Abdulla Eid b A College of Science . r D MATHS 103: Mathematics for Business I Dr. Abdulla Eid (University of Bahrain) Inverse Functions 1 / 8

d i E 1 Definition of inverse function. a l 2 Finding the inverse function. l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Inverse Functions 2 / 8

1- Definition of inverse function Recall: If a is a number, then a − a = 0 = − a + a . − a is an inverse of a d with respect to the addition +. i E If a is a non-zero number, then a 1 a = 1 = 1 a a . 1 a is an inverse of a a with respect to the multiplication · . l l u If f is a function, we want to find an “ inverse “ g to f with respect to d the composite ◦ , i.e., we want to find g (which is called the inverse) b A such that ( f ◦ g )( x ) = x and ( g ◦ f )( x ) = x . r D usually, we denote it by f − 1 . If f is a “nice“ function, we want to find an “inverse“ g Note: Not every function has an inverse! (we will see the horizontal line test later). Dr. Abdulla Eid (University of Bahrain) Inverse Functions 3 / 8

Finding the inverse function Step 0: Write y = f ( x ) . Step 1: Exchange x and y in step 0. Step 2: Solve the literal equation in step 1 for y (see Section 0.7). d Example i E (Old Exam Question) Find the inverse of g ( x ) = 5 x − 3. a l l Solution: Step 0: Write y = g ( x ) . u d b y = 5 x − 3 A Step 1: Exchange x and y in step 0. . r D x = 5 y − 3 Step 2: Solve the literal equation in step 1 for y x = 5 y − 3 x + 3 = 5 y Dr. Abdulla Eid (University of Bahrain) Inverse Functions 4 / 8

Continue... Step 2: Solve the literal equation in step 1 for y d i E x = 5 y − 3 a l x + 3 = 5 y l u x + 3 d = y b 5 A Hence we have . r g − 1 ( x ) = x + 3 D 5 Dr. Abdulla Eid (University of Bahrain) Inverse Functions 5 / 8

To check you answer, we have to check that g ( g − 1 ( x )) = x and g − 1 ( g ( x )) = x . 1 g ( g − 1 ( x )) = 5 ( g − 1 ( x )) − 3 � x + 3 d � = − 3 i E 5 a = ( x + 3 ) − 3 l l u = x d b 2 A g − 1 ( g ( x )) = g ( x ) + 3 . r D 5 = ( 5 x − 3 ) + 3 5 = 5 x 5 = x Dr. Abdulla Eid (University of Bahrain) Inverse Functions 6 / 8

Exercise Find the inverse of F ( x ) = ( 4 x − 5 ) 2 . d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Inverse Functions 7 / 8

Exercise Find the inverse of y = 3 2 x + 7 5 . d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Inverse Functions 8 / 8