91Ó°ÊÓ

In mathematics, a one-to-one function (also known as an injective function) is a function where every unique input maps to a unique output. This means that if you take any two distinct elements from the domain of the function, their images in the codomain must also be distinct.
If a function passes this test, it means that no two different inputs will produce the same output. This is crucial for determining if an inverse function can exist.
For our function, let’s analyze:\[f(x) = 3x + 1\]
We assume that \(f(a) = f(b)\) and see if this implies \(a = b\):
\[3a + 1 = 3b + 1\]
Subtracting 1 from both sides, we get:
\[3a = 3b\]
Dividing both sides by 3 gives:
\[a = b\]
This confirms that the function is one-to-one.

Once we establish that a function is one-to-one, the next step is to find its inverse. The inverse function essentially reverses the roles of the inputs and outputs.
To find the inverse of a function:\(f(x)\), follow these steps:
1. Replace \(f(x)\) with \(y\):
\[y = 3x + 1\]
2. Swap the roles of \(x\) and \(y\):
\[x = 3y + 1\]
3. Solve for \(y\) in terms of \(x\):
Subtract 1 from both sides:
\[x - 1 = 3y\]
Divide both sides by 3:
\[y = \frac{x - 1}{3}\]
This new equation gives us the inverse function:
\[f^{-1}(x) = \frac{x - 1}{3}\].
Remember, the inverse function undoes the action of the original function.

Understanding the properties of functions helps to comprehend what makes them one-to-one and how we find their inverses.
Some key properties include:

• Injective (One-to-One): Each element in the domain maps to a unique element in the codomain. We've verified this for \(f(x) = 3x + 1\).
• Inverse: A function must be one-to-one to have an inverse. The inverse function \(f^{-1}(x)\) reverses the output operation of \(f(x)\).
• Composite Function: A function composed with its inverse returns the original input. If we test \(f(f^{-1}(x))\), it should return \(x\).

Knowing these properties solidifies understanding and ensures correct usage when working with functions and their inverses.