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.
Visualise how mapping reverses, why the line \(y = x\) acts as an axis of symmetry, and how coordinates swap between \(f\) and \(f^{-1}\).
Understand the concept, condition, properties and algebraic method of inverse functions.
If \(f: x \to y\) or \(y = f(x)\), then the inverse function \(f^{-1}: y \to x\) or \(x = f^{-1}(y)\). Equivalently, \(y = f(x) \iff x = f^{-1}(y)\).
The superscript \(-1\) in \(f^{-1}\) represents the inverse function and does not mean a reciprocal: \(f^{-1}(x) \neq \dfrac{1}{f(x)}\).
A function \(f\) has an inverse function \(f^{-1}\) if and only if \(f\) is a one-to-one function. Many-to-one functions do not have inverse functions.
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.
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.
Domain of \(f^{-1}\) = Range of \(f\), and Range of \(f^{-1}\) = Domain of \(f\).
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 \(y = x\).
Set \(y = f(x)\), rearrange to express \(x\) as a function of \(y\), then replace \(y\) with \(x\) to obtain \(f^{-1}(x)\).
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
The inverse reverses the original mapping.
Composite relation for inverse functions
Both compositions must return the identity \(x\).
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 \(x\).
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.
Objective: Test your understanding of inverse function notation, conditions, properties and algebraic methods.
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?
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 ________.
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.
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