Algebra II Notes Inverse Functions Unit 1. Inverse o a Linear Function Math Background Previously, you Perormed operations with linear unctions Identiied the domain and range o linear unctions In this unit you will Solve a linear equation or a given y value Solve a literal equation or a given variable Find the inverse o a linear unction Describe how the domain o a unction must be restricted so the unction has an inverse You can use the skills in this unit to Find the inverse o a linear unction and recognize restrictions on its domain Solve or the independent variable in terms o the dependent variable in real-world situations Read values o an inverse unction rom a graph or a table, given that the unction has an inverse Graph the inverse o a unction by relecting the graph over the line y x Vocabulary Domain The input values o a relation. Horizontal line test The inverse o a unction is also a unction i and only i no horizontal line intersects the graph o more than once. Inverse unctions Two unctions are inverse unctions i the domain o the original unction matches the range o the second unction. Inverse relation Interchanges the input and output values o the original relation. Literal equation An equation that involves two or more variables. One-to-one unction A unction whose inverse is a unction. Both must pass the vertical and horizontal line tests Range The output values o a relation. Vertical line test A relation is a unction i and only i no vertical line intersects the graph o the unction more than once. Essential Questions How is the inverse o a relation related to the relation? How do you ind the inverse relation o a given unction? Overall Big Ideas The inverse o a relation is created by relecting the ordered pairs about the line y x. Relations can be restricted to orce the inverse to be a unction. Using Algebra we can ind the inverse unction or any line. Alg II Notes Unit 1. Inverse Functions Page 1 o 6 8/1/01
Algebra II Notes Inverse Functions Unit 1. Skill To ind and graph the inverse o a linear unction. Related Standards F.BF.B.a-1 Solve an equation o the orm cor simple linear and quadratic unctions that have an inverse and ( x + 1) write an expression or the inverse. For example, x or or x 1 ( x 1) Alg II Notes Unit 1. Inverse Functions Page o 6 8/1/01
Algebra II Notes Inverse Functions Unit 1. Notes, Examples, and Exam Questions In this unit, we will ind the inverse o a linear unction and see how the original unction is related to its inverse. Let s take a look at a mapping diagram. The one on the right top shows our original unction,. The mapping on the bottom shows the inverse o, The notation 1 1, the original outputs become the inputs, and the original inputs become the outputs. 1 indicates the inverse o a unction ( x ). The domain o is the range o ( x ), and the range o 1 is the domain o ( x ). **Note: The symbol -1 in 1 is not to be interpreted as an exponent. In other words,. 1 1 The procedure or inding the inverse o a linear unction is airly basic. 1) Substitute y or (x) ) Switch the x and the y in the equation ) Solve or y. You have now solved or the inverse o the unction. Ex 1 Find the ordered pairs or the unction and then ind the ordered pairs or the inverse unction. Show the inverse mapping on the igure below as well. Ans: Function: {(, -), (6, 0), (5,-6), (-, ), (-6, 5)} Inverse: {(-, ), (0, 6), -6, 5), (, -), (5, -6)} Inverse mapping is in pink. Alg II Notes Unit 1. Inverse Functions Page o 6 8/1/01
Algebra II Notes Inverse Functions Unit 1. Ex Find the inverse o x. I you need to ind the domain and range, look at the original unction and its graph and since it is a linear unction, the domain was all real numbers and the range is all real numbers. To ind the domain and range o the inverse, just swap the domain and range rom the original unction. For linear unctions, domain and range will always be all real numbers or the original unction and the inverse unction. Ex Here s the original unction: Interchange the x and the y: Solve or y: Rewrite in unction notation: y x x y x+ y x + y 1 x + Find the inverse o the given unction. x+ y x+ switch the variables and solve 1 x y+ x y ( x ) y Ans: x + y x The graph o an inverse relation is a relection o the graph o the original relation. The line o relection is y x. Graph the original unction, graph the line y x, and relect the igure over that line. The original relation is the set o ordered pairs: {(-, 1), (-1, ), (0, 0), (1, -), (, -)}. The inverse relation is the set o ordered pairs: {(1, -), (, -1), (0, 0), (-, 1), (-, )}. Notice that or the inverse relation the domain (x) and the range (y) reverse positions. Alg II Notes Unit 1. Inverse Functions Page o 6 8/1/01
Algebra II Notes Inverse Functions Unit 1. I no vertical line intersects the graph o a unction more than once, then is a unction. This is called the vertical line test. I no horizontal line intersects the graph o a unction more than once, then the inverse o is itsel a unction. This is called the horizontal line test. A unction is a one-to-one unction i and only i each second element corresponds to one and only one irst element. In order or the inverse o a unction to be a unction, the original unction must be a one-to-one unction and meet the criteria or the vertical and horizontal line tests. Ex Graph the inverse o 5 x+ 1. To graph the unction, three ordered pairs were ound: (0, 1), (, 6), and (-, -). The original unction is in blue. The red dashed line is the relection line y x and the ordered pairs were then relected over this line: (1, 0), (6, ), (-, -). The pink line represents the inverse graph o (x). Ex 5 The ormula F ( C) 1.8C+ converts temperatures in degrees Celsius, C, to degrees Fahrenheit, F. a) What is the input to the unction? What is the output? Ans: input: temperature in degrees Celsius, output: temperature in degrees Fahrenheit b) Find a ormula or the inverse unction giving Celsius as a unction o Fahrenheit. F F 1.8C+ F 1.8 C C 1.8 ***Note that you do not switch the variables when you are inding inverses o models. This would be conusing because the letters are chosen to remind you o the real-lie quantities they represent. c) Use inverse notation to write your ormula. d) What is the input to the unction -1? the output? 1 F ( F) 1.8 Ans: input: temperature in degrees Fahrenheit, output: temperature in degrees Celsius e) Interpret the meaning o the notation: (50) 1 Ans: 50 Celsius is 1 Fahrenheit ) Interpret the meaning o the notation: 1 (00) 9. Ans: 00 Fahrenheit is 9. Celsius Alg II Notes Unit 1. Inverse Functions Page 5 o 6 8/1/01
Algebra II Notes Inverse Functions Unit 1. Sample Exam questions 1. What is the inverse o x+ 9? A. B. 1 x + 9 C. 1 1 x + 9 D. x 1 9 1 Ans: C x 9. I A. B. 1 x+ 8, what is ( x )? ( x 8) C. x 8 D. x 6 ( x 8) Ans: A. I the point (a, b) lines on the graph y (x), the graph o y -1 (x) must contain point A. (0, b) B. (a, 0) C. (b, a) D. (-a, -b) Ans: B. The inverse unction o {(, 6), (-, ), (7, -5)} is A. {(-,6),(,),(-7,-5)} C. {(6,),(,-),(-5,7)} B. {(,-6),(-,-),(7,5)} D. {(-6,-),(-,),(5,-7)} Ans: C 5. Given the relation A: {(, ), (5, ), (6, ), (7,)} A. Both A and A -1 are unctions. C. Only A -1 is a unction. B. Neither A nor A -1 are unctions. D. Only A is a unction. Ans: D 6. The inverse o the unction x+ y 6 is A. B. y x+ C. y x+ D. y x+ y x+ Ans: B Alg II Notes Unit 1. Inverse Functions Page 6 o 6 8/1/01