# Function Inverses (HL) | Maths with Lemon

Source: https://mathswithlemon.io/aa-hl-function-inverses.html


### Function Inverses — HL

IB Mathematics: Analysis and Approaches · AHL 2.14 . Explore inverse functions, one-to-one restrictions, self-inverse functions, and odd and even functions. This lesson follows DPAAHL Lesson 2.8 .

Lesson 2.8 worksheet (PDF) AA Functions overview HL Function Transformations


#### 1. What Is an Inverse Function?


##### What you need to know

- Function notation and domain and range.
- The horizontal line test for one-to-one functions.
- Reflection in the line \(y=x\).

##### Key Points

An inverse function reverses the action of the original function. If \(f(a)=b\), then \(f^{-1}(b)=a\).

The domain and range exchange roles when an inverse is formed. The graph of \(f^{-1}\) is the reflection of the graph of \(f\) in \(y=x\).

Let \(f(x)=2x+3\). Set \(y=2x+3\), exchange \(x\) and \(y\), and solve:

Check: \(f^{-1}(f(x))=x\).


#### 2. Restricting Domains


##### From the worksheet

- Begin with \(f(x)=x^2+1\) on page 1.
- Practise quadratic and quartic restrictions on page 2.
- Sketch each function and its inverse on the same axes.

##### Key Points

For a function to have an inverse function , it must be one-to-one on its chosen domain. A quadratic needs a domain restriction to one side of its turning point.

Let \(f(x)=x^2+1\), with restricted domain \(x\geq0\).

The domain of \(f^{-1}\) is the range of \(f\), namely \([1,\infty)\).


#### 3. Interactive Inverse Graph Explorer


##### Explore the reflection

- Select a one-to-one function.
- Compare its graph with the inverse.
- Observe symmetry in the line \(y=x\).
- Notice how domain restrictions affect the inverse.
Blue: original function · Orange: inverse · Dashed: \(y=x\).


##### Compare functions


#### 4. Self-Inverse Functions


##### What you need to know

- A self-inverse function is its own inverse.
- Its graph is symmetric about \(y=x\).
- Verify by composition or algebraic rearrangement.
See the two self-inverse examples on page 1 of your worksheet.


##### Key Points

Subtract 3, interchange the variables, and solve:

Therefore \(f^{-1}(x)=f(x)\). Its graph is unchanged by reflection in \(y=x\).

Try: Show that \(f(x)=\dfrac{k}{x-1}+1\), for \(x>1\) and \(k>0\), is self-inverse.


#### 5. Odd and Even Functions


##### What you need to know

- Use \(f(-x)\) to test algebraically.
- Recognize symmetry about the \(y\)-axis and origin.
- Check that the domain is symmetric about zero.
Work through the tables and classifications on page 3.


##### Key Points

Symmetric about the \(y\)-axis. Example: \(f(x)=x^2\).

Symmetric about the origin. Example: \(f(x)=x^3\).

For \(f(x)=2x^3+3x\):

Therefore \(f\) is odd.


#### 6. HL Proofs and Practice


##### Worksheet challenges

- Page 4, Problem 1: choosing a domain for a quadratic inverse.
- Problem 2: finding a maximal one-to-one interval for a cubic.
- Problems 3–4: proofs involving odd and even functions.
- Problem 5: finding a parameter in a self-inverse function.

##### Revision checklist

Be ready to justify domain restrictions, find an inverse algebraically, state the inverse domain and range, identify self-inverse functions, and prove odd/even properties.

Then \(E(-x)=E(x)\), \(O(-x)=-O(x)\), and \(f(x)=E(x)+O(x)\), provided the domain is symmetric about zero.

Open the complete worksheet


### IB Past Paper Problems

Select a topic and unit, then click the button below to get a new problem.

Press the button to start!

How was this problem?