Question

Find the inverse function of the function \(f\) given by the set of ordered pairs. \(\\{(-1,1),(-2,2),(-3,3),(-4,4)\\}\)

Short answer

The function \(f^{-1}\), the inverse of \(f\), is given by the set of ordered pairs \{(1,-1),(2,-2),(3,-3),(4,-4)\}.

Step-by-step solution

Understanding the Function

Analyze the function. The function \(f\) is represented as ordered pairs: \((-1,1),(-2,2),(-3,3),(-4,4)\). Generally in these pairs, the first number is the 'x' value, and the second number is the 'y' value or 'f(x)', the value of the function at x.

Find the Inverse

The inverse function essentially swaps the coordinates of the x and y values. Therefore, to find the inverse function, swap the 'x' and 'y' coordinates for each pair in the given function. This results in ((1, -1), (2, -2), (3, -3), (4, -4)).

Verify the Inverse Function

Verify the new set of pairs derived in Step 2. An intuitive way to verify is the fact that if \(f(f^{-1}(x)) = x\) for every 'x' in function \(f^{-1}\), and \(f^{-1}(f(x)) = x\) for every 'x' in function 'f', then, \(f^{-1}\) is indeed the inverse.

Key concepts

Ordered Pairs

An ordered pair is a fundamental concept in mathematics and represents two elements. These elements are usually numbers and are shown within parentheses like this:

• The first element is known as the "x-coordinate."
• The second element is referred to as the "y-coordinate."

The order in which these numbers are written is crucial as it dictates their roles. In the context of functions, ordered pairs like \((-1, 1)\) indicate that the input or domain value \((x)\) is -1, and the corresponding output or range value \((y)\) is 1.
Ordered pairs are the building blocks for understanding functions, allowing us to see a direct relationship between the two elements. This makes it easier to analyze and understand how a particular function behaves.

X and Y Coordinates

Understanding x and y coordinates is key to working with functions.

• The x-coordinate, often referred to as the "input," is the first number in an ordered pair. It represents the horizontal position on a graph.
• The y-coordinate, the second number in the pair, is known as the "output" or the result of function evaluation for its corresponding x-coordinate, and represents vertical positioning on a graph.

In the function expressed as \((-1, 1), (-2, 2), (-3, 3), (-4, 4)\), each pair is a snapshot indicating that when \(x = -1\), \(y = 1\), and so on.
To find the inverse function, we swap these coordinates, essentially flipping the role of x and y. This helps in understanding how transformations and inverses work in a set context.

Function Verification

Verifying a function involves ensuring you have correctly identified the inverse.
To verify that we have the correct inverse of a function, we employ two main tests:

• First, for every \(x\) in the inverse function, pass it through the original function and check if you get \(x\) back: \(f(f^{-1}(x)) = x\).
• Second, for every \(x\) in the original function, go through the inverse and ensure \(x\) is returned: \(f^{-1}(f(x)) = x\).

This verification process ensures the inverse correctly reverses the function mappings, confirming its accuracy.
Thus, using this step-by-step checking, we ensure that switching the x and y coordinates produces the correct inverse function every time.