Composite & Inverse Functions (DP IB Applications & Interpretation (AI)): Revision Note

Composite Functions

What is a composite function?

• A composite function is where a function is applied to another function

• A composite function can be denoted

  • 

  • 

  • 

• The order matters

  • means:

    • First apply g to x to get 

    • Then apply f to the previous output to get 

    • Always start with the function closest to the variable

  • is not usually equal to 

How do I find the domain and range of a composite function?

• The domain of is the set of values of ...

  • which are a subset of the domain of g

  • which maps g to a value that is in the domain of f

• The range of  is the set of values of ...

  • which are a subset of the range of f

  • found by applying f to the range of g

• To find the domain and range of 

  • First find the range of g

  • Restrict these values to the values that are within the domain of f

    • The domain is the set of values that produce the restricted range of g

    • The range is the set of values that are produced using the restricted range of g as the domain for f

• For example: let  and 

  • The range of g is

    • Restricting this to fit the domain of f results in 

  • The domain of  is therefore 

    • These are the values of x which map to

  • The range of  is therefore 

    • These are the values which f maps to

Inverse Functions

What is an inverse function?

• Only one-to-one functions have inverses

• A function has an inverse if its graph passes the horizontal line test

  • Any horizontal line will intersect with the graph at most once

• The identity function  maps each value to itself

  • 

• If and  have the same effect as the identity function then  and  are inverses

• Given a function we denote the inverse function as 

• An inverse function reverses the effect of a function

  • means 

• Inverse functions are used to solve equations

  • The solution of  is 

• A composite function made of  and  has the same effect as the identity function

  • 

What are the connections between a function and its inverse function?

• The domain of a function becomes the range of its inverse

• The range of a function becomes the domain of its inverse

• The graph of  is a reflection of the graph in the line 

  • Therefore solutions to  or  will also be solutions to 

    • There could be other solutions to  that don't lie on the line 

How do I find the inverse of a function?

• STEP 1: Swap the x and y in 

  • If then 

• STEP 2: Rearrange  to make  the subject

• Note this can be done in any order

  • Rearrange  to make  the subject

  • Swap and 

Can many-to-one functions ever have inverses?

• You can restrict the domain of a many-to-one function so that it has an inverse

• Choose a subset of the domain where the function is one-to-one

  • The inverse will be determined by the restricted domain

  • Note that a many-to-one function can only have an inverse if its domain is restricted first

• For quadratics – use the vertex as the upper or lower bound for the restricted domain

  • For restrict the domain so 0 is either the maximum or minimum value

    • For example: or 

  • For restrict the domain so h is either the maximum or minimum value

    • For example: or 

• For trigonometric functions – use part of a cycle as the restricted domain

  • For restrict the domain to half a cycle between a maximum and a minimum

    • For example: 

  • For restrict the domain to half a cycle between maximum and a minimum

    • For example: 

  • For restrict the domain to one cycle between two asymptotes

    • For example: 

How do I find the inverse function after restricting the domain?

• The range of the inverse is the same as the restricted domain of the original function

• The inverse function is determined by the restricted domain

  • Restricting the domain differently will change the inverse function

• Use the range of the inverse to help find the inverse function

  • Restricting the domain of to means the range of the inverse is 

    • Therefore 

  • Restricting the domain of to means the range of the inverse is 

    • Therefore