Math 22 Functions
Basic Definitions
A function is a relationship between two variables such that to each value of the independent variable there corresponds exactly one value of the dependent variable.
The domain of the function is the set of all values of the independent variable for which the function is defined.
The range of the function is the set of all values taken on by the dependent variable.
Function notation: We usually denote a function f of x as . For example, function can be written as in function notation.
Exercises Find the domain and range of the following functions:
1)
Solution: |
---|
The domain is where the function defines (or all possible values of x). So, the radicand (everything under the square root) need to be non-negative. |
So, |
Answer: or |
The range is all of possible outcomes (values of y). Notice that is never negative. So is never negative. |
Answer: or |
Evaluate a Function
To evaluate a function at . We just need to plug in to find .
Example: Find the value of the function at
Answer:
Exercises Find the value of the function at the given values:
2) at
Solution: |
---|
isn't in the domain of . So, undefined |
OR |
Combinations of Functions
Two functions can be combine in varuious way. For example, let and . Then,
Composite Function
Let and be functions. The function given by is the composite function of and .
Examples: Let and
So,
Exercises Given and . Find each composite function below
1)
Solution: |
---|
2)
Solution: |
---|
Inverse Functions
Informally, the inverse function of is another function that “undoes” what has done. We usually denote as
Formal definition of inverse function. Let and be functions such that and Under these conditions, the function is the inverse function of , we denote
Important: The domain of must be equal to the range of , and the range of must be equal to the domain of
Exercise:
1) Show two functions and are inverses
Solution: |
---|
We want to show that these two functions satisfy and . So |
Consider |
and |
Hence, and are inverses |
2) Show two functions and are inverses
Solution: |
---|
We want to show that these two functions satisfy and . So |
Consider |
and |
Hence, and are inverses |
Finding Inverse Function
To find the inverse function of a given function . We can follow these steps: 1) Replace with 2) Interchange and 3) Solve for 4) Replace by
Exercises Find the inverse function of
1)
Solution: |
---|
Step 1: |
Step 2: |
Step 3: |
Step 4: |
2)
Solution: |
---|
Step 1: |
Step 2: |
Step 3: |
Step 4: |
This page were made by Tri Phan