Boundless Lecture Slides Free to share, print, make copies and changes. Get yours at Available on the Boundless Teaching Platform

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 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

Boundless is an innovative technology company making education more affordable and accessible for students everywhere. The company creates the world’s best open educational content in 20+ subjects that align to more than 1,000 popular college textbooks. Boundless integrates learning technology into all its premium books to help students study more efficiently at a fraction of the cost of traditional textbooks. The company also empowers educators to engage their students more effectively through customizable books and intuitive teaching tools as part of the Boundless Teaching Platform. More than 2 million learners access Boundless free and premium content each month across the company’s wide distribution platforms, including its website, iOS apps, Kindle books, and iBooks. To get started learning or teaching with Boundless, visit boundless.com.boundless.com Free to share, print, make copies and changes. Get yours at About Boundless

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 ect&utm_source=boundless

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 ?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

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 ?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

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 ?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

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 ?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

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 ?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

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 ?campaign_content=book_196_section_35&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Exponents and Logarithms > Inverse Functions

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 inverses ?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

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 composite-functions ?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

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 composite-functions ?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

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 composite-functions ?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

[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 ?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

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 ?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

[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 ?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

Free to share, print, make copies and changes. Get yours at Appendix

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 Exponents and Logarithms

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 Boundless. "Interactive Graph: Inverse Relationships." CC BY-SA View on Boundless.comCC BY-SA 3.0https:// on Boundless.com Exponents and Logarithms

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 Wikipedia. "Logarithm." GNU FDL View on Boundless.comGNU FDLhttp://en.wikipedia.org/wiki/Logarithm#Inverse_functionView on Boundless.com Exponents and Logarithms

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 Boundless. "Interactive Graph: Graph of a Parabola." CC BY-SA View on Boundless.comCC BY-SA 3.0https:// on Boundless.com Exponents and Logarithms

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 Boundless. "Interactive Graph: One-to-One Function and Its Inverse." CC BY-SA its-inverse View on Boundless.comCC BY-SA 3.0https:// its-inverseView on Boundless.com Exponents and Logarithms

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 Boundless. "Boundless." CC BY-SA View on Boundless.comCC BY-SAhttp:// on Boundless.com Exponents and Logarithms

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 Boundless. "Interactive Graph: Logarithm Function." CC BY-SA View on Boundless.comCC BY-SA 3.0https:// on Boundless.com Exponents and Logarithms

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 Wikipedia. "Composition of functions." GNU FDL View on Boundless.comGNU FDLhttp://en.wikipedia.org/wiki/Composition_of_functionsView on Boundless.com Exponents and Logarithms

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

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

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

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

Free to share, print, make copies and changes. Get yours at 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.

Free to share, print, make copies and changes. Get yours at Boundless - LO. "Boundless." CC BY-SA BY-SA 3.0http:// 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.

Free to share, print, make copies and changes. Get yours at 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|

Free to share, print, make copies and changes. Get yours at Boundless - LO. "Boundless." CC BY-SA BY-SA 3.0http:// 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|

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

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

Free to share, print, make copies and changes. Get yours at 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)

Free to share, print, make copies and changes. Get yours at Boundless - LO. "Boundless." CC BY-SA BY-SA 3.0http:// 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)

Free to share, print, make copies and changes. Get yours at 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

Free to share, print, make copies and changes. Get yours at Boundless - LO. "Boundless." CC BY-SA BY-SA 3.0http:// 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

Free to share, print, make copies and changes. Get yours at 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

Free to share, print, make copies and changes. Get yours at Boundless - LO. "Boundless." CC BY-SA BY-SA 3.0http:// 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

Free to share, print, make copies and changes. Get yours at 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.

Free to share, print, make copies and changes. Get yours at Boundless - LO. "Boundless." CC BY-SA BY-SA 3.0http:// 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.

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