Problem 20 - College Algebra with Modeling and Visualization

Question

The table is a complete representation of \(f\). Use the table to determine if \(f\) is one-to-one and has an inverse. \(\begin{array}{rrrrr}x & -2 & 0 & 2 & 4 \\ f(x) & 4 & 2 & 0 & -2\end{array}\)

Step-by-Step Solution

Answer

The function \( f \) is one-to-one and has an inverse.

Step 1: Understand the Definition of One-to-One Function

A function \( f \) is one-to-one if and only if each input (\( x \)) has exactly one unique output (\( f(x) \)) and vice versa. No two distinct inputs can map to the same output.

Step 2: Analyze the Table for Distinct Outputs

Observe the values of \( f(x) \) for each input \( x \) in the given table: \( 4, 2, 0, -2 \). Each output is unique with respect to its corresponding input, meaning no two different \( x \) values produce the same \( f(x) \).

Step 3: Determine One-to-One Correspondence

Since each output value is unique and corresponds solely to its input value, \( f \) is one-to-one. This suggests that each \( f(x) \) value is paired with exactly one \( x \) value and vice versa.

Step 4: Infer Existence of an Inverse Function

A function must be one-to-one to have an inverse. As established in previous steps, \( f \) is one-to-one, indicating that \( f \) has an inverse. For each \( f(x) \), there exists a unique \( x \) because \( f \) forms a bijection between its domain and range.

Key Concepts

Inverse Function

An inverse function essentially reverses the roles of inputs and outputs in the original function. If a function \(f\) is one-to-one, it means that each output comes from exactly one input value. Therefore, we can "undo" the function by finding out the original input for a given output. In mathematical terms, if \(f(x) = y\), then the inverse function, denoted as \(f^{-1}(y) = x\), gives back this original input.

The key condition here is that \(f\) must be one-to-one. Otherwise, two different inputs could produce the same output, leading to confusion about which input to invert back to. This property guarantees a perfect pairing between inputs and outputs, necessary for constructing an inverse. In our exercise, because each output value corresponds to a distinct input value, we can confidently say that \(f\) has an inverse function.

Function Table

A function table is a handy tool that helps visualize the correspondence between inputs and outputs. Think of it as a ledger that records what each \(x\) (input) becomes as it is transformed by the function into \(f(x)\) (output). This makes it easier to spot patterns and properties like one-to-one mappings.

In our specific table, each row shows a unique mapping:

• For \(x = -2\), \(f(x) = 4\)
• For \(x = 0\), \(f(x) = 2\)
• For \(x = 2\), \(f(x) = 0\)
• For \(x = 4\), \(f(x) = -2\)

This clear layout makes it easy to examine if each output is paired with just one input, which is crucial for determining if a function is one-to-one.

Unique Outputs

The uniqueness of outputs is a characteristic feature of one-to-one functions. Essentially, when each input \(x\) is mapped to a different output \(f(x)\), the function fulfills the criterion of being one-to-one.

In our case, the function \(f\) is derived from a well-organized table that distinctly assigns outputs to inputs without repetition. This lack of repeated outputs can be quickly verified: each output (4, 2, 0, -2) appears only once. Consequently, no different inputs share the same output. This distinctive quality not only verifies the one-to-one nature of \(f\) but also ensures that an inverse function can be established. It's all about preserving these unique relationships to maintain functionality both ways.