April 13, 2024

Rearranging functions

2.2.1 Inverse functions, f −1(x)

Inverse functions are the reverse of a function. Finding the input x for the output y. You can think of it as going backwards through the number machine.

Inverse functions

This is the same as reflecting a graph in the y = x axis.

Inverse functions 1

Finding the inverse function.

   f(x) = 2x3 + 3, find f −1(x)

  1. Replace f(x) with y

     y = 2x3 + 3

2. Solve for x

Inverse functions 2
3. Replace x with f −1(x) and y with x
Inverse functions 3

2.2.2 Composite functions

   Composite functions are a combination of two functions.

   (f ◦ g)(x) means f of g of x

To find the composite function above substitute the function of g(x) into the x of f (x).

Example: Let f(x) = 2x + 3 and g(x) = x2. Find (f ◦ g)(x) and (g ◦ f )(x).

Composite functions

Note: Remember f ◦ g(x) ≠ g ◦ f(x)

2.2.3 Transforming functions

By adding and/or multiplying by constants we can transform a function into another function.

Transforming functions

Exam hint: describe the transformation with words as well to guarantee marks. 

Always do translations last

Transforming functions f (x) → af (x + b)

Given f(x)=¼ x3 + x25/x, draw 3f (x−1).

  1. Sketch f(x)
Transforming functions 2

2. Stretch the graph by the factor of a

Transforming functions 3

3. Move graph by −b

Transforming functions 4