Problem 18 - Algebra for College Students

Question

(a) list the domain and range of the function, (b) form the inverse function \(f^{-1}\), and (c) list the domain and range of \(f^{-1}\). $$ f=\\{(-1,1),(-2,4),(-3,9),(-4,16)\\} $$

Step-by-Step Solution

Answer

(a) Domain: \([-1, -2, -3, -4]\), Range: \([1, 4, 9, 16]\); (b) \(f^{-1} = \{(1, -1), (4, -2), (9, -3), (16, -4)\}\); (c) Domain of \(f^{-1}\): \([1, 4, 9, 16]\), Range: \([-1, -2, -3, -4]\).

Step 1: Identify the Domain and Range of f

The domain of a function is the set of all possible input values (x-values), and the range is the set of all possible output values (y-values). For the given function \(f = \{(-1, 1), (-2, 4), (-3, 9), (-4, 16)\}\), the domain is \([-1, -2, -3, -4]\) and the range is \([1, 4, 9, 16]\).

Step 2: Form the Inverse Function

To find the inverse of the function \(f\), we swap the domain and range. The inverse function \(f^{-1}\) is thus \{(1, -1), (4, -2), (9, -3), (16, -4)\} because we interchanged each x-coordinate and y-coordinate pair from the original set.

Step 3: Identify the Domain and Range of the Inverse Function

The domain of the inverse function \(f^{-1}\) is the range of the original function \(f\), and the range of the inverse function \(f^{-1}\) is the domain of the original function \(f\). Therefore, the domain of \(f^{-1}\) is \([1, 4, 9, 16]\) and the range is \([-1, -2, -3, -4]\).

Key Concepts

Domain and Range

When we talk about a function's domain, we refer to all the possible input values, or x-values, that the function can accept. Similarly, the range encompasses all possible output values, or y-values, that the function can yield. For example, in the function\[f=\{(-1,1),(-2,4),(-3,9),(-4,16)\},\]the domain includes the numbers \[-1, -2, -3, \text{ and } -4\].These are the x-values you see in each pair. On the other hand, the range consists of values \[1, 4, 9, \text{ and } 16,\]which are the y-values in each pair.
Understanding domain and range sets us up perfectly for discussing other functions, such as inverse functions. Observing these values helps us track what happens when we manipulate a function.

Inverse Function

An inverse function essentially "undoes" the work of the original function. To find an inverse function, you simply swap each x-value with its corresponding y-value from the function's set of ordered pairs. If we have \[f=\{(-1,1),(-2,4),(-3,9),(-4,16)\},\]its inverse \[f^{-1}\]is\[(1,-1), (4,-2), (9,-3), (16,-4).\]
This swap means, for instance, if the original function maps \(-1\) to \(1\), the inverse function maps \(1\) back to \(-1\).
Thus, an inverse function retraces the steps of the original function. It is important to ensure that the function is one-to-one before finding its inverse; otherwise, the inverse won't be a function.

Coordinate Transformation

Coordinate transformation is a key concept when dealing with functions and their inverses. When you solve for an inverse function like \(f^{-1}\),you are engaging in a transformation of the coordinates. This involves interchanging x- and y-values, which can be thought of as a sort of flipping the graph over the line \(y=x\).
This process is not limited to just points in algebra but is a fundamental aspect of various mathematical analyses, such as physics, engineering, and even computer graphics.

• The inversion swaps coordinates, transforming the graph while preserving the relationships within the data points.
• Remember that these transformations require the original function to be one-to-one to ensure that the inverse is valid.

Coordinate transformation doesn't just help us find inverses but also provides insights into symmetries and relationships in different mathematical fields and applications.