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Exponential Function - Raleigh Charter High School指数函数-罗利特许高中
Unit 11.1 Inverse Functions
We 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 let’s 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):
Give an example of a function with a limited domain. What is it’s limitation?: 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.
Let’s 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 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 . Notice that once again we are using notation that looks like something else. You’re 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 isn’t, then it
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