Exponential, Logarithmic and Inverse Functions

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1 Chapter Review Sec.1 and. Eponential, Logarithmic and Inverse Functions I. Review o Inverrse I Functti ions A. Identiying One-to-One Functions is one-to-one i every element in the range corresponds to only one element in the domain. I a then a or i a then a A unction Horizontal Line Test: I there is NO horizontal line that intersects the graph more than once, then the unction is one-to-one. Eample: Determine whether each unction is one-to-one. 1.).) g B. Inverse Functions Let e a unction that is one-to-one and that goes through the point a, 1 Then is the inverse o 1 1 will go through the point, a 1 The domain o = the range o 1 The domain o = the range o C. Finding Inverse Functions Steps: 1. Test to see whether the unction is one-to-one. Replace with y. Interchange and y 4. Solve equation or y 1. Replace y with Eample: Veriy that the unctions are inverse o each other 1.) 4 and g 4 1

graph o over the line y Eamples: 1 1.) Construct the graph o i =.

2 Eample: Find the inverse or each o the ollowing k 1 1.).) g 1.) h D. Graphs o Inverse Functions The graph o 1 can e constructed y mirroring the graph o over the line y Eamples: 1 1.) Construct the graph o i =.) The ollowing are points on the graph o :, 10, 1,, 0,, 1, 1,, 6 1 Find at least points on the graph o

3 D. Domain and Range o Inverse Functions The domain o The domain o Eample: A unction 1 = the range o 1 = the range o, and 1 has the ollowing graph. Find the domain and Range o the inverse unction III I.. Review o Eponentti iall and Logarri itthmi ic Functti ions A. Eponential Functions DEFN: An eponential unction is a unction in the orm a. (i.e. the variale is in the eponent) Eample: Find or each o the ollowing: 1.) 4.) 7.) 4 B. Logarithmic Functions I. Logarithmic Functions A logarithm is a unction that helps us to solve a quadratic unction / logarithms allow us to isolate the variale in a quadratic unction (and the other way around). DEFN: A logarithmic unction is a unction in the orm log y = log a. (i.e. the variale is in the epression) y is equal to log ase o - Here is the BASE NUMBER and is the VARIABLE. log = y means eactly the same thing as y =

4 Eamples: Write each equation in its equivalent orm: 1.) log 16.) y = log 6 16.) log 7 4.) 8 y 00 II. Common Logarithmic Properties 1. log 1. log 1 0. log 0 DNE 4. log log. 6. log log 10 Eample: Simpliy Each Epression 1.) log.) log 6 1.) log ) log z z y.) log 8 1y 8 6.) log 10 III. The Natural Logarithm DEFN: e is a numer the equals approimately log e ln Eample: log.) ln e 1.) z e 4

5 IV. Epansion Properties or Logarithms 1. log M N log M log N (Product Rule). M log log M log N (Quotient Rule) N. n log M n log M (Power rule) Eample: Simpliy the ollowing: 1.) z log.) log 7.) log y 4.) log 10 y Epand the ollowing logarithms.) log y 6.) log y y 7.) log z 8.) ln e 9.) ln 4

6 ) log z 4 Write the ollowing as single logarithms 11.) log z 4log 1 1.) log log 1 log 1.) log log z y z 6

7 V. Change o Base Formula or Logarithms log M log M log ln M ln Eample: Use your calculator to ind: 1.) log 8 17.) log 1 C. Solving Eponential and Logarithmic Functions I. Common Base Property or Eponential Functions I M N, then M N Eample: 1.) Solve.) Solve 8 4 II. Eponentiating (How to solve equations involving e and ln) ln ln e ln Eample: 7 1.) Find i: e.) Find i: 1 7

8 III. Common Base Property or Logarithmic Functions I log M log N, then M N 1 ln Eample: Solve ln ln 4 IV. Solving or a variale in the eponent. Eample: 1.) R e 8.) y 8 e t 8

D. Graphs o Eponential and Logarithmic Functions I.

two graphs, showing the inversion line in red.

and ase numer) and are inverses o each other, the domain o one is the range o the other and vice

9 D. Graphs o Eponential and Logarithmic Functions I. Comparison o Logarithmic unction graph to Eponential unction graph y = y = log () Comparison o the two graphs, showing the inversion line in red. I a and g log then and g * Note: Since eponential and logarithmic unctions (with the same variale and ase numer) and are inverses o each other, the domain o one is the range o the other and vice versa. Eample: Find the domain and range o 1.) log.) 4 a.) ln 1 log are inverses o each other. a Domain: All (No Restrictions), Range: y > 0 0, log a Domain: > 0 0, Range: All (No Restrictions), 9

Math-3 Lesson 8-5. Unit 4 review: a) Compositions of functions. b) Linear combinations of functions. c) Inverse Functions. d) Quadratic Inequalities

Math-3 Lesson 8-5. Unit 4 review: a) Compositions of functions. b) Linear combinations of functions. c) Inverse Functions. d) Quadratic Inequalities Math- Lesson 8-5 Unit 4 review: a) Compositions o unctions b) Linear combinations o unctions c) Inverse Functions d) Quadratic Inequalities e) Rational Inequalities 1. Is the ollowing relation a unction