Composition of functions and inverse functions Goals: 1. Increase - - PowerPoint PPT Presentation

Composition of functions and inverse functions

Goals:

• 1. Increase fluency in the notation of inverse functions.
• 2. Make use of the definition of inverse functions in solving

problems.

• 3. Learn how to compute the inverse of linear functions.

Prepare:

• 1. Turn off your cell phone and put it away.
• 2. Have your packet out.
• 3. Commit to engaging in mathematical discourse for the next

50 minutes.

Question

How do you eat an elepehant?

Composition of functions.

Can you represent the function h(x) = (2 − 3x)2 as a composition of two functions? Click in:

• 1. yes.
• 2. no

Composition of functions.

Can you represent the function h(x) = x as a composition of two functions? Click in:

• 1. yes.
• 2. no

Inverse function.

Definition

Let f : R → R be a function. If there is another function g : R → R such that f ◦ g(x) = g ◦ f(x) = x, then g is the inverse of f and we write g = f−1.

Inverse functions.

We saw yesterday that not every function has an inverse (ex: f : R → R, f(x) = x2).

Question

Does the function f : R → R given by the rule f(x) = x3 have an inverse? Think about making graph to help you investigate the question. Click in:

• 1. yes.
• 2. no.

Inverse functions.

Question

Let f : R → R and g : R → R be two functions. Assume that f is an invertible function and f−1(x) = 3x3. If we know that f ◦ g(x) = 3x + 4 what is g(x)?

Inverse functions.

Question (3.33)

Suppose that f and g both have an inverse. Show that if h = f ◦ g, then the inverse of h is g−1 ◦ f−1.

Inverse functions.

Question

Suppose that f and g both have an inverse. What is the inverse

• f g ◦ f?

Inverse functions.

Question

f : R → R is a function given by the following algebraic rule: f(x) = 4x + 5

• 1. What is f(4)?
• 2. What is f−1(21)?
• 3. What is f−1(12)?
• 4. What is f−1(7)?
• 5. What is f−1(100)?
• 6. What is f−1(b)?
• 7. What is f−1(x)?