2.5 Inverse Functions

Transcription

1 . Invere unction What i an Invere unction? An invere unction i a unction that undoe another unction. A an eample the unction add the number to the variable. onequentl, an invere unction uch a g that ubtract rom the variable would eentiall undo the irt unction a demontrated in the ollowing eample. Eample: Show that and g are invere. Solution: Evaluate the irt unction at ome value, uch a. Now evaluate the econd unction at the output value o the irt unction. g g g The output rom thi econd unction i the ame a the input rom the irt. onequentl, we ended up where we tarted. A tated previoul, the econd unction undid the irt. A a mapping, an invere look like thi. Eample: Determine i the ollowing unction are invere o each other. a. and g 1 b. and g Solution: Thee can both be determined b inpection. a. Ye b. Ye

2 One-to-One unction: In order or a unction to have an invere it mut be a one-to-one unction. A unction i one-to-one i ever -value ha a unique -value. Eample: Determine i the ollowing et o ordered pair are one-to-one.,,,,, Not one-to-one a. { } b. {,,,4,, } One-to-one c. {,,,4,, } Not one-to-one d. {,,,10,4, } Not a unction In order to be a one-to-one unction, it i o coure necear to be a unction, that i, ever -value mut have a unique -value. Eample: Do each o the ollowing mapping repreent a one-to-one unction?? Not a one-to-one unction One-to-one unction. Horizontal Line Tet: I a horizontal line croe the graph o a unction or relation at no more than one point, then the unction i one-to-one. The parabola to the right i not a one-to-one unction becaue it ail the horizontal line tet. Thi tet i imilar to the vertical line tet which i ued to determine i a relation i a unction.

3 Invere unction Notation: Given a unction, then it invere i it eit will be 1 and i read invere o. A an eample i then the invere o i 1. inding an Invere: onider the ollowing mapping o the unction and it invere 1. rom the mapping we ee that the input and output witch between the unction and it invere. I repreent the input variable and repreent the output variable, then we ma a that the and value witch. Eample: Given the unction {,,,,,1,4,6 } in et notation, ind it invere. Solution: To ind the invere o thi unction, witch the and value. {,,,,1,,6,4 } onequentl, inding an invere i impl a matter o witching the input and output value. Although ome invere unction ma be determined b inpection, mot cannot. Thereore it i necear to develop the ollowing procedure to ind invere unction. 1. Replace the unctional notation with a variable uch a.. Switch the variable.. Solve or the other variable. 4. Rewrite in unctional notation.

4 Eample: Given the unction, ind 1. Solution: Replace the unctional notation with the variable, olve or, then witch the variable and rewrite in unctional notation. 1 Eample: Given the unction, ind 1. Solution: Replace the unctional notation with the variable, olve or, then witch the variable and rewrite in unctional notation. 1 Eample: Given the unction, ind 1. Solution: Replace the unctional notation with the variable, olve or, then witch the variable and rewrite in unctional notation. 1

5 eriing an Invere: To veri that a unction i an invere o another unction we will ue a compoition o unction. Recall that a compoite unction i one unction that will do what two other unction will do. Once again, conider the mapping. Notice that the compoite unction take the input and create output that are the ame. In other word, when a compoite unction i created with a unction and it invere: Output Input Thereore, i the input i, the output will be. Thi lead u to the ollowing procedure or veriing invere. I two unction and 1 are invere o each other, then the ollowing i true. 1 and 1 Eample: eri that and 1 are invere. Solution: We need to veri that 1 and that Since 1 and 1, the unction are invere o each other.

6 Eample: eri that and 1 are invere. 1 Solution: We need to veri that and that Since 1 and 1, the unction are invere o each other. The Graph o an Invere: Since the invere o a unction ma be ound b witching the -value and -value input and output, then to graph the invere o a unction all we need to do i witch the value in a table. Eample: Graph the unction 1 and it invere b witching the -value and -value in a table. Solution: reate a table o value or the unction 1, and then witch the value to create the table or the invere. 1 X Graph both unction b plotting the ordered pair. The graph below alo demontrate an important relationhip between the graph o a unction and it invere, which i that the are relection over the line.

7 Eample: Given the unction 6, do the ollowing: 1. Show that it i one-to-one,. ind it invere,. eri the invere, 4. Graph the unction and it invere. Solution: 1. To determine i the unction 6 i one-to-one, we mut graph it and determine i it pae the horizontal line tet. The unction ma be quickl graphed uing tranormation. Since the unction pae the horizontal line tet, it i one-toone and thereore ha an invere.. ind the invere We need to veri that and that Graph the invere unction.

8 .-Application Application problem that ue invere uuall involve olving an equation or the independent variable. Thi i demontrated in the ollowing eample. Eample: The ta on a new car i % o the purchae price. Epre the total cot, including ta, a a unction o the purchae price. Epre the purchae price P a a unction o the total cot. eri that thee two unction are invere o each other. Solution: Let p equal the purchae price. Then 0.0p will repreent the ta on the car. The unction that epree cot a a unction o p i: p p 0.0 p p 1.0 p To epre the price p a a unction o, ind the invere unction. p 1.0 p 1.0 p p 1.0 c P c 1.0 To veri the invere we mut ubtitute each equation into the other and obtain a true tatement. When doing thi, do not ue unctional notation. 1.0P P P 1.0 P P P

9 Eample: The unction w 1. 46w epree the landing peed in eet per econd a a unction o the gro weight o the Lear. ind the invere unction and veri that it i the invere. Ue the invere unction to ind the gro weight or a lear or which the proper landing peed i 1 eet per econd. Solution: To ind the invere unction, olve the equation or w. w 1.46w 1.46w 1.46w w 1.46 To veri the invere we mut ubtitute each equation into the other and obtain a true tatement. 1.46w w w w w w 1.46 w w Ue the invere unction to ind the gro weight or a lear or which the proper landing peed i 1 eet per econd. Thereore the Lear weigh 10,444.1 lb. w w 1.46 w 10,444.1

10 Eample: The unction epree the eliu temperature a a unction o the ahrenheit temperature. ind the invere unction and then veri that it i the invere. What i it ued or? Solution: To ind the invere, olve the equation or. To veri the invere we mut ubtitute each equation into the other and obtain a true tatement. Thi unction epree the ahrenheit temperature a a unction o the eliu temperature. Eample: Epre the volume o a cube a a unction o the length o a ide. Then, epre the length o a ide o a cube a a unction o the volume. Solution: The volume o a cube a a unction o the length ma be written a: To epre the length o a ide o a cube S a a unction o the volume, olve the equation or and then write uing requeted notation.