College Algebra Chapter 4 Exponential and Logarithmic Functions

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Presentation transcript:

College Algebra Chapter 4 Exponential and Logarithmic Functions Section 4.1 Inverse Functions

1. Identify One-to-One Functions 2. Determine Whether Two Functions Are Inverses 3. Find the Inverse of a Function

Identify One-to-One Functions A function f is a one-to-one function if for a and b in the domain of f, or equivalently,

Example 1: Determine if the relation defines y as a one-to-one function of x.

Example 2: Determine if the relation defines y as a one-to-one function of x.

Identify One-to-One Functions A function y = f (x) is a one-to-one function if no horizontal line intersects the graph in more than one place.

Example 3: Determine if the relation defines y as a one-to-one function of x.

Example 4: Determine if the relation defines y as a one-to-one function of x.

Example 5: Determine if the relation defines y as a one-to-one function of x.

Example 6: Use the definition of a one-to-one function to determine whether the function is one-to-one. (Show that if )

Example 7: Use the definition of a one-to-one function to determine whether the function is one-to-one. (Show that if )

1. Identify One-to-One Functions 2. Determine Whether Two Functions Are Inverses 3. Find the Inverse of a Function

Determine Whether Two Functions Are Inverses Inverse Functions: Let f be a one-to-one function. Then g is the inverse of f if the following conditions are both true. Given a function and its inverse , then the definition implies that

Example 8: Determine whether the two functions are inverses.

Example 9: Determine whether the two functions are inverses.

1. Identify One-to-One Functions 2. Determine Whether Two Functions Are Inverses 3. Find the Inverse of a Function

Find the Inverse of a Function Procedure to Find an Equation of an Inverse of a Function For a one-to-one function defined by y = f (x), the equation of the inverse can be found as follows: Step 1 Replace f (x) by y. Step 2 Interchange x and y. Step 3 Solve for y. Step 4 Replace y by .

Example 10: A one-to-one function is given. Write an equation for the inverse function.

Example 10 continued:

Example 11: A one-to-one function is given. Write an equation for the inverse function.

Example 11 continued:

Example 11 continued:

Example 11 continued:

Example 11 continued:

Example 12: A one-to-one function is given. Write an equation for the inverse function.

Example 12 continued:

Example 13: The graph of a function is given. Graph the inverse function.