1.3 Inverse Functions
An inverse function reverses the mapping of an original function, returning an output value back to its original input. Written using the notation \(f^{-1}\), it operates such that if \(f(x) = y\), then \(f^{-1}(y) = x\). Not all functions have an inverse; an inverse function exists only if the original function is a one-to-one function. Graphically, an inverse function is the reflection of the original function along the line \(y = x\), which directly interchanges the domain and range as well as the coordinates of points on the graph.
Learning Objectives
- Describe and understand the concept and notation of an inverse function, \(f^{-1}\).
- State and verify the condition for the existence of an inverse function (one-to-one function) using discrete mappings and the horizontal line test.
- Verify whether two functions \(f\) and \(g\) are inverses of each other using the composite relation \(fg(x) = gf(x) = x\).
- Determine the properties of inverse functions concerning domain, range, coordinates \((a, b) \leftrightarrow (b, a)\), and line of reflection \(y = x\).
- Determine the algebraic formula of an inverse function \(f^{-1}(x)\) from a given function \(f(x)\).
Geometric and Algebraic Relationships of an Inverse Function
Visualise how mapping reverses, why the line \(y = x\) acts as an axis of symmetry, and how coordinates swap between \(f\) and \(f^{-1}\).
Short Explanation
Understand the concept, condition, properties and algebraic method of inverse functions.
1. Concept and Notation
If \(f: x \to y\) or \(y = f(x)\), then the inverse function \(f^{-1}: y \to x\) or \(x = f^{-1}(y)\). Equivalently, \(y = f(x) \iff x = f^{-1}(y)\).
2. Important Notation Note
The superscript \(-1\) in \(f^{-1}\) represents the inverse function and does not mean a reciprocal: \(f^{-1}(x) \neq \dfrac{1}{f(x)}\).
3. Condition for Existence
A function \(f\) has an inverse function \(f^{-1}\) if and only if \(f\) is a one-to-one function. Many-to-one functions do not have inverse functions.
4. Horizontal Line Test
If any horizontal line intersects the graph at only one point, the function is one-to-one and has an inverse. If it cuts at two or more points, there is no inverse function.
5. Composite Function Relationship
Two functions \(f\) and \(g\) are inverse functions of each other if and only if \(fg(x) = x\) and \(gf(x) = x\) over their respective domains.
6. Domain and Range Relationship
Domain of \(f^{-1}\) = Range of \(f\), and Range of \(f^{-1}\) = Domain of \(f\).
7. Coordinate Inversion and Reflection
If point \((a, b)\) lies on the graph of \(f\), then \((b, a)\) lies on the graph of \(f^{-1}\). The graph of \(f^{-1}\) is the reflection of the graph of \(f\) in the line \(y = x\).
8. Determining Inverse Algebraically
Set \(y = f(x)\), rearrange to express \(x\) as a function of \(y\), then replace \(y\) with \(x\) to obtain \(f^{-1}(x)\).
Steps for Finding the Inverse Function Algebraically
Change the equation \(y = f(x)\) to express \(x\) in terms of \(y\).
Write \(x\) as \(f^{-1}(y)\).
Replace the variable \(y\) with \(x\) to state the final inverse function \(f^{-1}(x)\).
State any domain restrictions, such as non-zero denominator conditions.
Inverse function equivalence
\(y = f(x) \iff x = f^{-1}(y)\)The inverse reverses the original mapping.
Composite relation for inverse functions
\(fg(x) = gf(x) = x\)Both compositions must return the identity \(x\).
Try to Answer First
Answer in your mind, then press "Check Answer".
A function is given as \(f(x) = 3x - 2\). Find the inverse function \(f^{-1}(x)\).
Determine whether the quadratic function \(f(x) = 4 - x^2\) for the domain of all real numbers has an inverse function. Give a reason.
Given that \(f(x) = \dfrac{5x}{x - 6}, x \neq 6\), find \(f^{-1}(x)\) and state the condition for \(x\).
Common Mistakes
- Confusing the inverse notation \(f^{-1}(x)\) with the reciprocal \(\dfrac{1}{f(x)}\) or \([f(x)]^{-1}\).
- Forgetting to state non-zero denominator conditions for rational inverse functions, such as writing \(f^{-1}(x) = \dfrac{6x}{x - 5}\) without specifying \(x \neq 5\).
- Failing to recognise that many-to-one functions, such as unrestricted quadratics \(y = ax^2 + bx + c\), do not have inverse functions.
- Mixing up the swap of variables: swapping \(x\) and \(y\) too early without algebraic justification or forgetting to rewrite \(f^{-1}(y)\) in terms of \(x\) at the final step.
- Assuming that \(fg(x) = x\) alone is sufficient to prove two functions are inverses, without also ensuring or checking that \(gf(x) = x\) over their respective domains.
Concept Misunderstandings
The power \(-1\) in \(f^{-1}(x)\) means \(\dfrac{1}{f(x)}\).
In function notation, the superscript \(-1\) denotes an inverse operation, not an algebraic exponent or index. \(f^{-1}(x) \neq \dfrac{1}{f(x)}\).
Every mathematical function automatically has an inverse function.
Only one-to-one functions have inverse functions. Many-to-one functions have reverse mappings that are relations, not functions, unless their domain is restricted.
The vertical line test is used to check if a function has an inverse.
The vertical line test tests whether a graph is a function. The horizontal line test tests whether a function is one-to-one and therefore has an inverse function.
Summary
- \(f: x \to y \iff f^{-1}: y \to x\) and \(y = f(x) \iff x = f^{-1}(y)\).
- \(f^{-1}(x)\) denotes the inverse function of \(f(x)\), and \(f^{-1}(x) \neq \dfrac{1}{f(x)}\).
- A function \(f\) has an inverse function \(f^{-1}\) if and only if \(f\) is a one-to-one function.
- The horizontal line test determines whether a function has an inverse: it must cut the curve at only one point.
- Composite relationship: \(f\) and \(g\) are inverses of each other if and only if \(fg(x) = x\) and \(gf(x) = x\).
- Domain and range interchange: Domain of \(f^{-1}\) = Range of \(f\), and Range of \(f^{-1}\) = Domain of \(f\).
- Reflection property: The graph of \(f^{-1}\) is the reflection of the graph of \(f\) in the line \(y = x\). If \((a, b)\) lies on \(f\), then \((b, a)\) lies on \(f^{-1}\).
- Algebraic method: Set \(y = f(x)\), rearrange to express \(x\) as a function of \(y\), and replace \(y\) with \(x\) to obtain \(f^{-1}(x)\).
Short Activity
Objective: Test your understanding of inverse function notation, conditions, properties and algebraic methods.
A. Objective Quiz
1 Which of the following statements is true regarding a function \(f\) and its inverse function \(f^{-1}\)?
2 Which test on a Cartesian graph determines whether a function \(f(x)\) has an inverse function?
B. Fill in the Blanks
3 Two functions \(f\) and \(g\) are inverse functions of each other if and only if \(fg(x) = gf(x) = \) ________.
4 If the point \((3, 7)\) lies on the graph of the one-to-one function \(f\), then the corresponding point on the graph of \(f^{-1}\) has coordinates ________.
C. Matching / Drag and Drop
Match each inverse function concept or property to its correct description or expression. On a phone, tap a term first, then tap the matching box.
