Problem 47 - A Graphical Approach to College Algebra

Question

Determine whether each function is one-to-one. If so, find its inverse. $$f=\left\\{\left(0,0^{2}\right),\left(1,1^{2}\right),\left(2,2^{2}\right),\left(3,3^{2}\right),\left(4,4^{2}\right)\right\\}$$

Step-by-Step Solution

Answer

The function is one-to-one; its inverse is \(\{(0,0), (1,1), (4,2), (9,3), (16,4)\}\).

Step 1: Identify Function Type

The given function is expressed as a set of ordered pairs: \(\{(0,0), (1,1), (2,4), (3,9), (4,16)\}\). It is a finite function defined on the set of integers.

Step 2: Check for One-to-One Property

A function is one-to-one if each input corresponds to a unique output and no output is repeated. For this set of ordered pairs, each unique \(x\)-value (input) has a unique \(y\)-value (output), and no \(y\) is repeated, so the function is one-to-one.

Step 3: Finding the Inverse

Since the function is one-to-one, its inverse can be found by swapping each \(x\) and \(y\) in the ordered pairs. The inverse thus becomes \(\{(0,0), (1,1), (4,2), (9,3), (16,4)\}\).

Key Concepts

One-to-One Functions

A one-to-one function ensures that every input has a unique output. This means no two different inputs can map to the same output.
Validating if a function is one-to-one involves checking its set of ordered pairs. Here, each pair \((x, y)\) must have distinct \(y\)-values:

• If \(x_1 eq x_2\), then \(f(x_1) eq f(x_2)\).
• Each \(y\) value is used only once across the function.

For the function \(f=\{(0,0), (1,1), (2,4), (3,9), (4,16)\}\):

• 0 corresponds uniquely to 0.
• 1 corresponds uniquely to 1.
• 2 corresponds uniquely to 4.
• 3 corresponds uniquely to 9.
• 4 corresponds uniquely to 16.

Since all \(y\) values are unique, the function is one-to-one. This allows us to find its inverse.

Ordered Pairs

Every function can be seen as a collection of ordered pairs. These pairs are written as \((x, y)\), where \(x\) is the input and \(y\) is the output.
For functions, they help define the relationship between variables:

• Each ordered pair represents a link, connecting each \(x\) to exactly one \(y\).
• The list of \((x, y)\) pairs gives a complete picture of the function’s behavior.

Ordered pairs are foundational for determining if functions are one-to-one.
With the example \(\{(0,0), (1,1), (2,4), (3,9), (4,16)\}\), we directly map inputs to their outputs, confirming the function's properties and enabling us to find an inverse.

Finite Function

A finite function has a limited set of ordered pairs, meaning it doesn’t go on forever.
These are easier to analyze and understand because the scope is small and defined:

• The functions are restricted to a fixed number of inputs and outputs.
• Allows for complete evaluation without applying theoretical limits.

In our example, \(f=\{(0,0), (1,1), (2,4), (3,9), (4,16)\}\), the function is finite because it only has five pairs.
This finite nature simplifies checking if it’s one-to-one, and finding its inverse is straightforward by swapping each \(x\) and \(y\) in the pairs.