Inverse Function

Definition

Let $f: A \to B$ be a function. The inverse function of $f$, denoted by $f^{-1}$, is a function from $B$ to $A$ such that:

$$f^{-1}(f(x)) = x \quad \text{for all } x \in A$$

and

$$f(f^{-1}(y)) = y \quad \text{for all } y \in B$$

In simple words, applying a function and then applying its inverse brings you back to the original value, and vice versa.

Important conditions for existence

• A function must be one-to-one (injective) so that different inputs do not produce the same output.
• A function must be onto (surjective) if the inverse is to be a function from the entire codomain back to the domain.
• Therefore, a function must be bijective to have an inverse function on the full codomain.

Example:

If $f(x)=3x-5$, then its inverse exists because the function is one-to-one and onto over real numbers.

To find it: $$y=3x-5$$ $$x=3y-5$$ $$y=\frac{x+5}{3}$$

So, $$f^{-1}(x)=\frac{x+5}{3}$$

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Main Content

1. One-to-One and Onto Nature of Inverse Functions

• A function can have an inverse function only when it is bijective, meaning it is both one-to-one and onto.

One-to-one (injective)

• means each output is produced by exactly one input. If two different inputs give the same output, the inverse will not be a function because one output would map to multiple inputs.

Onto (surjective)

• means every element of the codomain is reached by the function. If some codomain elements are never produced, the inverse cannot be defined for those values unless the codomain is restricted.

Example:

Consider $f(x)=x^2$ with domain and codomain as real numbers.

• It is not one-to-one because $f(2)=4$ and $f(-2)=4$.
• So, its inverse is not a function over all real numbers.

But if the domain is restricted to $x \ge 0$, then $f(x)=x^2$ becomes one-to-one, and the inverse is: $$f^{-1}(x)=\sqrt{x}$$

Relationship with relations:

In relation theory, inverse relation of a relation $R\subseteq A\times B$ is obtained by swapping ordered pairs: $$R^{-1}=\{(b,a)\mid (a,b)\in R\}$$ For functions, the inverse function must also satisfy the function property, not just be an inverse relation.

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2. Graphical Interpretation and Reflection Property

• The graph of a function and the graph of its inverse are mirror images of each other about the line: $$y=x$$

• This happens because the inverse swaps the roles of $x$ and $y$. If $(a,b)$ lies on the graph of $f$, then $(b,a)$ lies on the graph of $f^{-1}$.

• This graphical property is very useful for checking whether an inverse has been correctly found.

Example:

If a point on $f$ is $(1,4)$, then the corresponding point on $f^{-1}$ is $(4,1)$.

Simple visual representation:

            y
            |
      (4,1) |        •
            |      /
            |    /   y = x
            |  /
------------+---------------- x
            |  \
            |    \ 
            |      \
      (1,4) |        •

This shows that points on one graph switch coordinates on the inverse graph.

Important idea:

If the graph of a function fails the horizontal line test, it is not one-to-one, and therefore it does not have an inverse function over that domain.

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3. Methods for Finding and Verifying the Inverse

• The standard method for finding the inverse of a function is to:
• Replace $f(x)$ by $y$,
• Swap $x$ and $y$,
• Solve for $y$,
• Write the result as $f^{-1}(x)$.

• After finding the inverse, it should be verified using composition: $$f(f^{-1}(x))=x \quad \text{and} \quad f^{-1}(f(x))=x$$

• This verification confirms that the two functions truly undo each other.

Example 1: Linear function

Let $$f(x)=2x+7$$

Step 1: $$y=2x+7$$

Step 2: interchange $x$ and $y$ $$x=2y+7$$

Step 3: solve for $y$ $$x-7=2y$$ $$y=\frac{x-7}{2}$$

So, $$f^{-1}(x)=\frac{x-7}{2}$$

Verification: $$f(f^{-1}(x))=2\left(\frac{x-7}{2}\right)+7=x-7+7=x$$

Example 2: Rational function

Let $$f(x)=\frac{1}{x}$$ where $x\neq 0$.

The inverse is: $$f^{-1}(x)=\frac{1}{x}$$

This function is its own inverse, meaning applying it twice returns the original value.

Special note:

Not every algebraic expression leads to a valid inverse over the same domain. Sometimes the domain or codomain must be restricted so that the inverse becomes a function.

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Working / Process

1. Check whether the function is invertible

• Determine whether the function is one-to-one using algebraic reasoning or the horizontal line test.
• If necessary, restrict the domain to make the function one-to-one.
• Confirm that the function is onto the intended codomain if a complete inverse function is required.

2. Find the inverse algebraically

• Write $f(x)$ as $y$.
• Exchange $x$ and $y$.
• Solve for $y$ in terms of $x$.
• Rewrite the result as $f^{-1}(x)$.

3. Verify the result

• Compute $f(f^{-1}(x))$.
• Compute $f^{-1}(f(x))$.
• Both compositions should simplify to $x$, showing that the inverse is correct.
• Also check whether the domain and codomain are consistent with the inverse.

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Advantages / Applications

• Inverse functions help us reverse transformations, which is useful in solving equations and undoing operations.
• They are used in mathematical modeling, such as converting between units, decoding messages, and reversing formulas in science and engineering.
• Inverse functions are important in computer science and cryptography, where one process encrypts data and another inverse process decrypts it.
• They are also essential in calculus, especially when differentiating inverse trigonometric functions and solving advanced function-related problems.
• In set theory and relations, inverse functions provide a clear example of how bijection connects two sets in a reversible way.
• They support theorem proving techniques by allowing proof through composition, contradiction, and properties of injective and surjective mappings.

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Summary

• An inverse function reverses the action of a function and brings outputs back to original inputs.
• A function must be bijective to have an inverse on its full codomain.
• The inverse graph is a reflection of the original graph across the line $y=x$.