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2. Using Boundless Presentations The Appendix The appendix is for you to use to add depth and breadth to your lectures. You can simply drag and drop slides from the appendix into the main presentation to make for a richer lecture experience. Free to edit, share, and copy Feel free to edit, share, and make as many copies of the Boundless presentations as you like. We encourage you to take these presentations and make them your own. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless Teaching Platform Boundless empowers educators to engage their students with affordable, customizable textbooks and intuitive teaching tools. The free Boundless Teaching Platform gives educators the ability to customize textbooks in more than 20 subjects that align to hundreds of popular titles. Get started by using high quality Boundless books, or make switching to our platform easier by building from Boundless content pre-organized to match the assigned textbook. This platform gives educators the tools they need to assign readings and assessments, monitor student activity, and lead their classes with pre-made teaching resources. Get started now at: If you have any questions or problems please email: educators@boundless.com http://boundless.com/teaching-platform

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4. Inverses One-to-One Functions Finding Formulas for Inverses Composition and Composite Functions Restricting Domains Inverse Functions Exponents and Logarithms > Inverse Functions Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra?campaign_content=book_196_section_35&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=dir ect&utm_source=boundless

5. In trigonometric functions, a positive exponent applied to the function's abbreviation means that the result is raised to that power, while an exponent of −1 denotes the inverse function. An inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa (i.e., ƒ(x)=y, and g(y)=x). Inverses Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/exponents-and-logarithms-5/inverse-functions-35/inverses-165- 11071?campaign_content=book_196_section_35&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boun dless Logarithm Function View on Boundless.com Exponents and Logarithms > Inverse Functions

6. The logarithm to base b is the inverse function of f(x) = bx: [Equation 1]. Exponents and Logarithms > Inverse Functions Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/exponents-and-logarithms-5/inverse-functions-35/inverses-165- 11071?campaign_content=book_196_section_35&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boun dless Equation 1 View on Boundless.com

7. The natural logarithm ln(x) is the inverse of the exponential function ex:[Equation 2]. Exponents and Logarithms > Inverse Functions Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/exponents-and-logarithms-5/inverse-functions-35/inverses-165- 11071?campaign_content=book_196_section_35&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boun dless Equation 2 View on Boundless.com

8. A one-to-one function has an inverse function, if a function is not one-to-one it cannot have an inverse function as it yields multiple outputs. One-to-One Functions Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/exponents-and-logarithms-5/inverse-functions-35/one-to-one-functions-166- 5843?campaign_content=book_196_section_35&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess View on Boundless.com Exponents and Logarithms > Inverse Functions

9. Domain restriction can allow a function to become one-to-one, as in the case of [Equation 3]. Exponents and Logarithms > Inverse Functions Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/exponents-and-logarithms-5/inverse-functions-35/one-to-one-functions-166- 5843?campaign_content=book_196_section_35&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Equation 3 View on Boundless.com

10. An easy way to check if a function is a one-to-one function is by graphing it and then performing the horizontal line test. Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/exponents-and-logarithms-5/inverse-functions-35/one-to-one-functions-166- 5843?campaign_content=book_196_section_35&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Exponents and Logarithms > Inverse Functions

11. An inverse function reverses the inputs and outputs. To find the inverse formula of a function, write it in the form of y and x, switch y and x, and then solve for y. Some functions have no inverse function, as a function cannot have multiple outputs. Finding Formulas for Inverses Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/exponents-and-logarithms-5/inverse-functions-35/finding-formulas-for- inverses-167- 5873?campaign_content=book_196_section_35&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess View on Boundless.com Exponents and Logarithms > Inverse Functions

12. The composition of functions is always associative. That is, if f, g, and h are three functions with suitably chosen domains and codomains, then f ∘ (g ∘ h) = (f ∘ g) ∘ h, where the parentheses serve to indicate that composition is to be performed first for the parenthesized functions. Composition and Composite Functions Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/exponents-and-logarithms-5/inverse-functions-35/composition-and- composite-functions-168- 5885?campaign_content=book_196_section_35&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Composition of Functions View on Boundless.com Exponents and Logarithms > Inverse Functions

13. Functions can be inverted and then composed, giving the notation of: [Equation 4]. Exponents and Logarithms > Inverse Functions Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/exponents-and-logarithms-5/inverse-functions-35/composition-and- composite-functions-168- 5885?campaign_content=book_196_section_35&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Equation 4 View on Boundless.com

14. Functions can be composed and then inverted, yielding the following notation: [Equation 5]. Exponents and Logarithms > Inverse Functions Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/exponents-and-logarithms-5/inverse-functions-35/composition-and- composite-functions-168- 5885?campaign_content=book_196_section_35&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Equation 5 View on Boundless.com

15. [Equation 6] is defined as the inverse function of f(x) if it consistently reverses the f(x) process. Restricting Domains Exponents and Logarithms > Inverse Functions Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/exponents-and-logarithms-5/inverse-functions-35/restricting-domains-169- 5858?campaign_content=book_196_section_35&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Equation 6 View on Boundless.com

16. Informally, a restriction of a function f is the result of trimming its domain. Restricting Domains Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/exponents-and-logarithms-5/inverse-functions-35/restricting-domains-169- 5858?campaign_content=book_196_section_35&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess View on Boundless.com Exponents and Logarithms > Inverse Functions

17. [Equation 7],without any domain restriction, does not have an inverse function as it fails the horizontal line test. Exponents and Logarithms > Inverse Functions Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/exponents-and-logarithms-5/inverse-functions-35/restricting-domains-169- 5858?campaign_content=book_196_section_35&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Equation 7 View on Boundless.com

18. Free to share, print, make copies and changes. Get yours at www.boundless.com Appendix

19. Key terms composite function A function of one or more independent variables, at least one of which is itself a function of one or more other independent variables; a function of a function domain The set of points over which a function is defined. injective function In mathematics, an injective function or an injection is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain. inverse function A function that does exactly the opposite of another. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms

20. Interactive Graph: Inverse Relationships The graphs of (blue) and (red) are inverses of one another. The black line represents the line of reflection, in which is y=x. Try creating your own inverse. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph: Inverse Relationships." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-inverse-relationships View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-inverse-relationshipsView on Boundless.com Exponents and Logarithms

21. Logarithm Function The graph of the logarithm function logb(x) (blue) is obtained by reflecting the graph of the function bx (red) at the diagonal line (x = y). Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Logarithm." GNU FDL http://en.wikipedia.org/wiki/Logarithm#Inverse_function View on Boundless.comGNU FDLhttp://en.wikipedia.org/wiki/Logarithm#Inverse_functionView on Boundless.com Exponents and Logarithms

22. Interactive Graph: Graph of a Parabola Graph of a quadratic equation y=x2. This function forms a parabola and therefore fails the the horizontal line test. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph: Graph of a Parabola." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-graph-of-a-parabola View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-graph-of-a-parabolaView on Boundless.com Exponents and Logarithms

23. Interactive Graph: One-to-One Function and Its Inverse Graph of y=10x is a one-to-one function and is shown here with its inverse, y=log(x). Try making your own inverse relationship on a graph. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph: One-to-One Function and Its Inverse." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-one-to-one-function-and- its-inverse View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-one-to-one-function-and- its-inverseView on Boundless.com Exponents and Logarithms

24. Interactive Graph: Failure of Horizontal Line Test Graph of a parabola with the equation. This function fails the horizontal line test and therefore does not have an inverse. However, if we restrict the domain to be x>0 then we find that it passes the horizontal line test. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Boundless." CC BY-SA http://www.boundless.com/ View on Boundless.comCC BY-SAhttp://www.boundless.com/View on Boundless.com Exponents and Logarithms

25. Interactive Graph: Logarithm Function The graph of the logarithm function and. In this case, 5log(x) (blue) is obtained by reflecting the graph of the function 5x (red) at the diagonal line (x = y). Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph: Logarithm Function." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-logarithm-function View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-logarithm-functionView on Boundless.com Exponents and Logarithms

26. Composition of Functions g ∘ f, the composition of f and g. For example, (g ∘ f)(c) = #. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Composition of functions." GNU FDL http://en.wikipedia.org/wiki/Composition_of_functions View on Boundless.comGNU FDLhttp://en.wikipedia.org/wiki/Composition_of_functionsView on Boundless.com Exponents and Logarithms

27. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms Convert y=log35 to exponential form. A) 5y=3 B) y=35 C) 3y=5 D) y=53

28. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Exponents and Logarithms Convert y=log35 to exponential form. A) 5y=3 B) y=35 C) 3y=5 D) y=53

29. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms Covert 8=2x to logarithmic form. A) x=log82 B) 2=logx8 C) x=log28 D) 8=log2x

30. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Exponents and Logarithms Covert 8=2x to logarithmic form. A) x=log82 B) 2=logx8 C) x=log28 D) 8=log2x

31. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms A one-to-one function will: A) pass only the horizontal line test, but not the vertical line test. B) pass only the vertical line test, but not the horizontal line test. C) fail both the horizontal and vertical line tests. D) pass both the horizontal and vertical line tests.

32. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Exponents and Logarithms A one-to-one function will: A) pass only the horizontal line test, but not the vertical line test. B) pass only the vertical line test, but not the horizontal line test. C) fail both the horizontal and vertical line tests. D) pass both the horizontal and vertical line tests.

33. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms Which of the following functions is an example of a one-to-one function? A) f(x)=x2 B) f(x)=|x| C) f(x)=x3 D) f(x)=|x3|

34. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Exponents and Logarithms Which of the following functions is an example of a one-to-one function? A) f(x)=x2 B) f(x)=|x| C) f(x)=x3 D) f(x)=|x3|

35. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms What is the inverse of the function f(x)=3x? A) log3x B) logx3 C) log3x D) logx3

36. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Exponents and Logarithms What is the inverse of the function f(x)=3x? A) log3x B) logx3 C) log3x D) logx3

37. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms What is the inverse of f(x)=3x-5? A) y=5x-3 B) y=(3/x)+(3/5) C) y=(x/5)+(3/5) D) y=(x/3)+(5/3)

38. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Exponents and Logarithms What is the inverse of f(x)=3x-5? A) y=5x-3 B) y=(3/x)+(3/5) C) y=(x/5)+(3/5) D) y=(x/3)+(5/3)

39. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms Given f(x)=2x+4 and g(x)=7x+6, what is f(g(x))? A) 7(2x+4)+6 B) 9x+10 C) 27x+6+4 D) 4x+13

40. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Exponents and Logarithms Given f(x)=2x+4 and g(x)=7x+6, what is f(g(x))? A) 7(2x+4)+6 B) 9x+10 C) 27x+6+4 D) 4x+13

41. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms The inverse of the function y=x3 can only be found if its domain is restricted to: A) x>0 B) x>-1 C) No restriction is necessary. D) x<0

42. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Exponents and Logarithms The inverse of the function y=x3 can only be found if its domain is restricted to: A) x>0 B) x>-1 C) No restriction is necessary. D) x<0

43. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms The inverse of the function y=|x| can be found if its domain is restricted to: A) x>0 B) x>-1 C) x<1 D) No restriction is necessary.

44. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Exponents and Logarithms The inverse of the function y=|x| can be found if its domain is restricted to: A) x>0 B) x>-1 C) x<1 D) No restriction is necessary.

45. Attribution Wikipedia. "One-to-one function." CC BY-SA 3.0 http://en.wikipedia.org/wiki/One-to-one_functionCC BY-SA 3.0http://en.wikipedia.org/wiki/One-to-one_function Wikipedia. "Horizontal line test." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Horizontal_line_testCC BY-SA 3.0http://en.wikipedia.org/wiki/Horizontal_line_test Wikipedia. "injective function." CC BY-SA 3.0 http://en.wikipedia.org/wiki/injective%20functionCC BY-SA 3.0http://en.wikipedia.org/wiki/injective%20function Connexions. "Function Concepts -- Inverse Functions." CC BY 3.0 http://cnx.org/content/m18198/latest/CC BY 3.0http://cnx.org/content/m18198/latest/ Wikipedia. "Function (mathematics." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Function_(mathematicsCC BY-SA 3.0http://en.wikipedia.org/wiki/Function_(mathematics Wiktionary. "domain." CC BY-SA 3.0 http://en.wiktionary.org/wiki/domainCC BY-SA 3.0http://en.wiktionary.org/wiki/domain Connexions. "Function Concepts -- Inverse Functions." CC BY 3.0 http://cnx.org/content/m18198/latest/CC BY 3.0http://cnx.org/content/m18198/latest/ Wikipedia. "Inverse function." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Inverse_functionCC BY-SA 3.0http://en.wikipedia.org/wiki/Inverse_function Wiktionary. "inverse function." CC BY-SA 3.0 http://en.wiktionary.org/wiki/inverse+functionCC BY-SA 3.0http://en.wiktionary.org/wiki/inverse+function Wikipedia. "Composition of functions." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Composition_of_functionsCC BY-SA 3.0http://en.wikipedia.org/wiki/Composition_of_functions Wikipedia. "Inverse function." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Inverse_function#Inverses_and_compositionCC BY-SA 3.0http://en.wikipedia.org/wiki/Inverse_function#Inverses_and_composition Connexions. "Function Concepts -- Composite Functions." CC BY 3.0 http://cnx.org/content/m18187/latest/CC BY 3.0http://cnx.org/content/m18187/latest/ Wiktionary. "composite function." CC BY-SA 3.0 http://en.wiktionary.org/wiki/composite+functionCC BY-SA 3.0http://en.wiktionary.org/wiki/composite+function Wikipedia. "Exponentiation." CC BY-SA 3.0 http://en.wikipedia.org/wiki/ExponentiationCC BY-SA 3.0http://en.wikipedia.org/wiki/Exponentiation Wikipedia. "Logarithm." CC BY-SA 3.0 http://en.wikipedia.org/wiki/LogarithmCC BY-SA 3.0http://en.wikipedia.org/wiki/Logarithm Wikipedia. "Inverse functions." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Inverse_functionsCC BY-SA 3.0http://en.wikipedia.org/wiki/Inverse_functions Wiktionary. "inverse function." CC BY-SA 3.0 http://en.wiktionary.org/wiki/inverse+functionCC BY-SA 3.0http://en.wiktionary.org/wiki/inverse+function Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms