Inverse functions

In mathematics, one of the most fascinating concepts is that of inverse functions. If you’re new to the idea, don’t worry—this article will walk you through what they are, how they work, and why they’re important, using simple words and relatable examples.

What is an Inverse Function?

At its core, an inverse function undoes the work of a given function. Imagine a machine: you put something in, and the machine transforms it into something else. An inverse function would take the output of the machine and turn it back into the original input.

Definition on Inverse functions

A function f(x)f(x)f(x) has an inverse function f−1(x)f^{-1}(x)f−1(x) if, when you apply both functions one after the other, you get back to where you started. Mathematically:

f(f−1(x))=xandf−1(f(x))=xf(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x

f(f−1(x))=xandf−1(f(x))=x

This means that applying fff and then f−1f^{-1}f−1, or vice versa, cancels out the effect of the first function.

Breaking it Down with an Example

Let’s take a simple function:

f(x)=2xf(x) = 2x

f(x)=2x

This function doubles whatever you input. For instance:

• If you input 2, the output is f(2)=2×2=4f(2) = 2 \times 2 = 4f(2)=2×2=4.
• If you input 5, the output is f(5)=2×5=10f(5) = 2 \times 5 = 10f(5)=2×5=10.

Now, think about reversing this. If the output is 4, what input gives this result? To find it, we divide by 2:

f−1(x)=x2f^{-1}(x) = \frac{x}{2}

f−1(x)=2x

So:

• For f−1(4)=42=2f^{-1}(4) = \frac{4}{2} = 2f−1(4)=24=2
• For f−1(10)=102=5f^{-1}(10) = \frac{10}{2} = 5f−1(10)=210=5

Here, f(x)=2xf(x) = 2xf(x)=2x and f−1(x)=x2f^{-1}(x) = \frac{x}{2}f−1(x)=2x are inverses of each other. Together, they undo each other’s work.

Key Properties of Inverse Functions

1. Reversibility: Inverse functions "reverse" the original function. Applying the function and then its inverse (or vice versa) returns the original value:

f(f−1(x))=xandf−1(f(x))=xf(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = xf(f−1(x))=xandf−1(f(x))=x

How to Find the Inverse of a Function

Let’s walk through the steps to find the inverse of a function.

Step 1: Replace f(x)f(x)f(x) with yyy

If the function is given as f(x)=3x+5f(x) = 3x + 5f(x)=3x+5, rewrite it as:

y=3x+5y = 3x + 5

y=3x+5

Step 2: Swap xxx and yyy

Since the inverse switches the roles of xxx and yyy, replace xxx with yyy and yyy with xxx:

x=3y+5x = 3y + 5

x=3y+5

Step 3: Solve for yyy

Rearrange the equation to isolate yyy:

x−5=3y⇒y=x−53x - 5 = 3y \quad \Rightarrow \quad y = \frac{x - 5}{3}

x−5=3y⇒y=3x−5

Step 4: Replace yyy with f−1(x)f^{-1}(x)f−1(x)

The inverse function is:

f−1(x)=x−53f^{-1}(x) = \frac{x - 5}{3}

f−1(x)=3x−5

Applications of Inverse Functions

Inverse functions are used in many real-world scenarios and areas of mathematics. Here are some examples:

1. Solving Equations: Inverse functions help solve equations. For example, if f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, and you know f(x)=11f(x) = 11f(x)=11, you can use the inverse to find xxx.
2. Trigonometry: Inverse trigonometric functions (like sin⁡−1(x)\sin^{-1}(x)sin−1(x), cos⁡−1(x)\cos^{-1}(x)cos−1(x)) are used to find angles when given their sine or cosine values.
3. Physics and Engineering: Inverse functions are used in formulas that need to reverse calculations, such as finding time from speed and distance.
4. Cryptography: Many encryption algorithms rely on mathematical functions and their inverses to encode and decode information.

Special Cases: Inverse of Common Functions

1. Linear Functions:
   • If f(x)=ax+bf(x) = ax + bf(x)=ax+b, the inverse is f−1(x)=x−baf^{-1}(x) = \frac{x - b}{a}f−1(x)=ax−b.
2. Quadratic Functions:
   • Quadratics can have inverses only if you restrict the domain (e.g., f(x)=x2f(x) = x^2f(x)=x2 has an inverse only for x≥0x \geq 0x≥0).
   • The inverse is f−1(x)=xf^{-1}(x) = \sqrt{x}f−1(x)=x (or −x-\sqrt{x}−x for x<0x < 0x<0).
3. Exponential and Logarithmic Functions:
   • If f(x)=exf(x) = e^xf(x)=ex, then f−1(x)=ln⁡(x)f^{-1}(x) = \ln(x)f−1(x)=ln(x), because exponentials and logarithms are inverses of each other.
4. Trigonometric Functions:
   • For sin⁡(x)\sin(x)sin(x), the inverse sin⁡−1(x)\sin^{-1}(x)sin−1(x) gives the angle whose sine is xxx. Similar rules apply for cos⁡(x)\cos(x)cos(x), tan⁡(x)\tan(x)tan(x), etc.

Challenges and Misconceptions

1. Not Every Function Has an Inverse: Functions that fail the horizontal line test (where a horizontal line intersects the graph more than once) don’t have inverses because they’re not one-to-one.
2. Restricting the Domain: For some functions like f(x)=x2f(x) = x^2f(x)=x2, restricting the domain is necessary to make them one-to-one so that they can have an inverse.
3. Graph Interpretation: Many people struggle to visualize how functions and their inverses relate. Remember, their graphs are reflections across the line y=xy = xy=x.