Maths with Lemon

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 overviewHL 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 \(f^{-1}(b)=a\).

\[f^{-1}(f(x))=x\]

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 \(y=x\).

The symbol \(f^{-1}(x)\) means inverse function, not \(1/f(x)\).
Worked example: linear function

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

\[x=2y+3\quad\Longrightarrow\quad f^{-1}(x)=\frac{x-3}{2}\]

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.

Worked example: restricting a quadratic

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

\[y=x^2+1\quad\Longrightarrow\quad x=\sqrt{y-1}\]
\[f^{-1}(x)=\sqrt{x-1},\qquad x\geq1\]

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

Choosing \(x\leq0\) instead gives the different inverse branch \(f^{-1}(x)=-\sqrt{x-1}\).

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

\[f^{-1}(x)=f(x)\quad\Longleftrightarrow\quad f(f(x))=x\]
Worked example from Lesson 2.8
\[f(x)=\frac{5}{x-3}+3,\qquad x e3\]

Subtract 3, interchange the variables, and solve:

\[y-3=\frac{5}{x-3}\quad\Longrightarrow\quad x=\frac{5}{y-3}+3\]

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

Even functions
\[f(-x)=f(x)\]

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

Odd functions
\[f(-x)=-f(x)\]

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

Worked example

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

\[f(-x)=-2x^3-3x=-f(x)\]

Therefore \(f\) is odd.

Some functions are neither odd nor even. The zero function is both.

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.

HL proof: every function can be split into even and odd parts
\[E(x)=\frac{f(x)+f(-x)}{2}\]
\[O(x)=\frac{f(x)-f(-x)}{2}\]

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

Quick check: inverse of \(f(x)=(x-4)^2-2\), \(x\leq4\)
\[f^{-1}(x)=4-\sqrt{x+2},\qquad x\geq-2\]

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!