ࡱ> 8:7TKq` )bjbjqPqP 4\::!D (%(%(%8`%%<DRA4&&Z+:+++o,o,o,@@@@@@@$BhpDF@o,o,o,o,o,@++ A+/+/+/o,^++@+/o,@+/+/;h<+(& @ q(%,<@"A0RA%<E-*E <E<o,o,+/o,o,o,o,o,@@/o,o,o,RAo,o,o,o,DDD(%DDD(%$T Unit 11.1 Inverse Functions We are going to look at exponential functions and logarithmic functions in this unit. For example  EMBED Equation.DSMT4 is an exponential function and  EMBED Equation.DSMT4 is a logarithmic function. These functions relate to each other. They are inverses of each other. So lets look at inverse functions first. Review: Define domain: all the inputs, the x values Define range: all the outputs, the y values Give an example of a function with an unlimited domain (i.e. all real numbers):  EMBED Equation.DSMT4  Give an example of a function with a limited domain. What is its limitation?:  EMBED Equation.DSMT4 the number -2 is not in the domain as it makes the denominator zero and thus the function is undefined. What is the range of the two example functions: the range of the first function is all real numbers and the range of the second is all real numbers with the exception of zero. One-to-One functions: In a one-to-one function, each x-value corresponds to only one y-value, and each y-value corresponds to just one x-value. What is the difference between this definition and the definition of a function we talked about earlier this year? A function only requires that each x-value have one and only one y-value. A y-value may belong to more than one x-value. This is not true in a one-to-one function. Only one-to-one functions have an inverse. Lets define function  EMBED Equation.DSMT4  If we interchange the x and y values we get the inverse of function G.  EMBED Equation.DSMT4  Because we interchange the x and y values to get the inverse, it follows that the domain of the original function is the range of the inverse and the range of the original function becomes the domain of the inverse. The function notation for inverse functions is  EMBED Equation.DSMT4 . Notice that once again we are using notation that looks like something else. Youre just going to have to go by the context of the problem. In this context we are looking at an inverse function, not a negative exponent. To find an inverse function you first have to determine if the function is a one-to-one function. If it isnt, then it doesnt have an inverse and you can stop looking! Example:  EMBED Equation.DSMT4  This is not a one-to-one function because the y-value 2 corresponds to two x-values. No inverse.  EMBED Equation.DSMT4  is a on-to-one function and thus has an inverse  EMBED Equation.DSMT4  Horizontal Line Test: We use a Vertical Line Test to help us determine if a particular equation is a function. In a similar fashion we use a Horizontal Line Test to help us determine if a particular equation is a one-to-one function. If every horizontal line intersects the graph of the function at only one point then the function is a one-to-one function. (Note the equation has to first pass the vertical line test and be a function. Examples to put on board:     Finding the equation of the inverse of a function Rewrite the function notation as a y = form of equation. Interchange the y and the x. Solve for y. Rewrite in inverse function notation by replacing the y with  EMBED Equation.DSMT4  Example: Use the function from above  EMBED Equation.DSMT4 . Rewrite as y =  EMBED Equation.DSMT4  Interchange the y and the x  EMBED Equation.DSMT4  Solve for y  EMBED Equation.DSMT4  Rewrite in inverse function notation  EMBED Equation.DSMT4  Examples:  EMBED Equation.DSMT4  inverse is  EMBED Equation.DSMT4   EMBED Equation.DSMT4  inverse is  EMBED Equation.DSMT4   EMBED Equation.DSMT4  there is no inverse this is not a one-to-one function. Why not? An inverse function undoes its related function. One way to look at this is you will end up back where you started. For example, plug a number into the function in #1 above, take the output and plug it into the inverse. What do you get? Unit 11.2 Exponential Function For a > 0 and a ( 1 and all real numbers x,  EMBED Equation.DSMT4  defines the exponential function with base a. Graph  EMBED Equation.DSMT4  (blue line) and  EMBED Equation.DSMT4  (green line)  This graph is typical of exponential functions where a > 1. The larger the value of a the faster the graph rises. Graph  EMBED Equation.DSMT4  (blue line) and  EMBED Equation.DSMT4  (green line)  Here as x gets larger y gets smaller. Characteristics of the graph of  EMBED Equation.DSMT4  The graph contains the point (0, 1). If a > 1 the graph rises from left to right. If 0 < a < 1, the graph falls from left to right. Both go from 2nd to 1st quadrants. The x-axis is an asymptote. The domain is  EMBED Equation.DSMT4  and the range is  EMBED Equation.DSMT4 . What happens when you graph a more complicated exponential function such as  EMBED Equation.DSMT4 ? (Very similar to  EMBED Equation.DSMT4  except it is shifted to the right and rises more rapidly.) How would we solve  EMBED Equation.DSMT4 ? Property for solving an exponential Equation For a > 0 and a ( 1, if  EMBED Equation.DSMT4 . Solving an Exponential Equation Each side must have the same base. If the two sides do not have the same base, express each as a power of the same base. Simplify exponents, if necessary, using the rules of exponents. Set exponents equal using the exponential equation property. Solve the equation from Step 3. (Note: for equations such as  EMBED Equation.DSMT4  we need to use a different method because Step 1 is difficult to do.) Example:  EMBED Equation.DSMT4  Examples:  EMBED Equation.DSMT4  Unit 11.3 Logarithmic Functions What is a logarithm? A logarithm is an exponent. Remember in the previous unit I said that we couldnt solve  EMBED Equation.DSMT4  because it is difficult to express both sides of the equation as expressions with the same base? Well, there are ways to use logarithms to solve equations such as this. (Weve got a ways to go before were ready for that. (check page 695)). Logarithm For all positive numbers a, with a ( 1, and all positive numbers x,  EMBED Equation.DSMT4 . This is saying the expression  EMBED Equation.DSMT4  represents the exponent to which the base a must be raised in order to get x.  EMBED Equation.DSMT4  Solving logarithmic equations  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  For any positive real number b, with b ( 1,  EMBED Equation.DSMT4  Unit 11.4 Properties of Logarithms Product Rule for Logarithms  EMBED Equation.3  The logarithm of a product is equal to the sum of the logarithms of the factors. Proof of the Product Rule  EMBED Equation.DSMT4  Examples:  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Quotient Rule for Logarithms If x, y, and b are positive real numbers, where b ( 1, then  EMBED Equation.DSMT4  The logarithm of a quotient is the difference between the logarithm of the numerator and the logarithm of the denominator. Examples:  EMBED Equation.DSMT4   EMBED Equation.DSMT4  ? any restrictions on x?  EMBED Equation.DSMT4  this illustrates an important point. When solving logs, if you can convert a log to a number then do so. Power Rule for Logarithms If x and b are positive real numbers, where b ( 1, and if r is any real number, then  EMBED Equation.DSMT4  Recall this example from the Product Rule:  EMBED Equation.DSMT4  Examples:  EMBED Equation.DSMT4   EMBED Equation.DSMT4  when using the power rule with logarithms of expressions involving radicals, first rewrite the radical expression with a rational exponent.  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Special Properties of Logarithms If b > 0 and b ( 1, then  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  The second is the easiest to understand. To prove the first property: Let  EMBED Equation.DSMT4  Properties of Logarithms Product Rule  EMBED Equation.3  Quotient Rule  EMBED Equation.DSMT4  Power Rule  EMBED Equation.DSMT4  Special Properties  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  Writing Logarithms in Alternative Forms  EMBED Equation.DSMT4   EMBED Equation.DSMT4      Unit 11 Exponential and Logarithmic Functions  FILENAME Unit 11 - Student Notes Page  PAGE 1 of  NUMPAGES 8 y   * + , - } ~  1 B [ ﻮ~na~j0 hjUshjUs6Uj&Q hjUshjUs6UVjhjUshjUs6UjhjUshjUsEHUj&Q hjUshjUsUVhjUshjUs6j^hjUshjUsEHUj&Q hjUshjUsUVjhjUshjUsEHUj&Q hjUshjUsUVjhjUshjUsU hjUshjUshjUshjUs6>*%a b j .  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