Exponential Function - Raleigh Charter High School指数函数-罗利特许高中

1、Unit 11 Exponential and Logarithmic FunctionsUnit 11.1 Inverse FunctionsWe are going to look at exponential functions and logarithmic functions in this unit. For example is an exponential function and is a logarithmic function.These functions relate to each other. They are inverses of each other. So

2、 lets look at inverse functions first.Review:Define domain: all the inputs, the x valuesDefine range: all the outputs, the y valuesGive an example of a function with an unlimited domain (i.e. all real numbers): Give an example of a function with a limited domain. What is its limitation?: the number

3、-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-t

4、o-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.

5、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 If we interchange the x and y values we get the inverse of function G.Because we interchange the x and y values to get the inverse, it follows tha

6、t 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 . Notice that once again we are using notation that looks like something else. Youre just going to have to go by t

7、he 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:This is not a one-to

8、-one function because the y-value 2 corresponds to two x-values. No inverse. is a on-to-one function and thus has an inverseHorizontal 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 dete

9、rmine 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 equ

10、ation of the inverse of a function1. Rewrite the function notation as a “y =” form of equation.2. Interchange the y and the x.3. Solve for y.4. Rewrite in inverse function notation by replacing the y with Example: Use the function from above . 1. Rewrite as “y =” 2. Interchange the y and the x3. Sol

11、ve for y 4. Rewrite in inverse function notationExamples: 1. inverse is 2. inverse is 3. 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 int

12、o the function in #1 above, take the output and plug it into the inverse. What do you get?Unit 11.2 Exponential FunctionFor a > 0 and a ¹ 1 and all real numbers x, defines the exponential function with base a.Graph (blue line) and (green line)This graph is typical of exponential functions wh

13、ere a > 1. The larger the value of a the faster the graph rises.Graph (blue line) and (green line)Here as x gets larger y gets smaller.Characteristics of the graph of 1. The graph contains the point (0, 1).2. If a > 1 the graph rises from left to right. If 0 < a < 1, the graph falls from

14、 left to right. Both go from 2nd to 1st quadrants.3. The x-axis is an asymptote.4. The domain is and the range is .What happens when you graph a more complicated exponential function such as ? (Very similar to except it is shifted to the right and rises more rapidly.)How would we solve ?Property for

15、 solving an exponential EquationFor a > 0 and a ¹ 1, if .Solving an Exponential Equation1. 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.2. Simplify exponents, if necessary, using the rules of exponents.3. Set exponent

16、s equal using the exponential equation property.4. Solve the equation from Step 3.(Note: for equations such as we need to use a different method because Step 1 is difficult to do.)Example:Examples: Unit 11.3 Logarithmic FunctionsWhat is a logarithm?A logarithm is an exponent. Remember in the previou

17、s unit I said that we couldnt solve 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).LogarithmFor all positive n

18、umbers a, with a ¹ 1, and all positive numbers x, . This is saying the expression represents the exponent to which the base a must be raised in order to get x.Solving logarithmic equations For any positive real number b, with b ¹ 1,Unit 11.4 Properties of LogarithmsProduct Rule for Logarit

19、hms The logarithm of a product is equal to the sum of the logarithms of the factors.Proof of the Product RuleExamples:Quotient Rule for LogarithmsIf x, y, and b are positive real numbers, where b ¹ 1, then The logarithm of a quotient is the difference between the logarithm of the numerator and the logarithm of the denominator.Examples: ? any restrictions on x? this illustrates an important point. When solving logs, if you can con