Problem 1 - Algebra A Combined Function

Question

Determine whether each function is a one-to-one function. If it is one-to-one, list the inverse function by switching coordinates, or inputs and outputs. \(g=\\{(0,3),(3,7),(6,7),(-2,-2)\\}\)

Step-by-Step Solution

Answer

The function is not one-to-one, so it does not have an inverse.

Step 1: Understanding the Problem

We are given a function represented as a set of ordered pairs: \(g=\{(0,3),(3,7),(6,7),(-2,-2)\}\). We need to determine whether this function is one-to-one. A one-to-one function has no repetition of the output values.

Step 2: Check for Output Repetition

To determine if the function is one-to-one, check if any y-value is repeated in the set of ordered pairs. The function \(g\) has the ordered pairs: \( (0,3), (3,7), (6,7), (-2,-2)\). Notice that the y-value 7 is repeated twice, at points (3,7) and (6,7).

Step 3: Conclusion: Not One-to-One

Since the y-value 7 is repeated for different x-values (3 and 6), the function \(g\) is not a one-to-one function. A one-to-one function cannot have duplicate outputs for different inputs.

Step 4: Inverse Check

Since the function is not one-to-one, it cannot have an inverse function that is also a function. For an inverse to exist as a function, each output value of \(g\) must map to exactly one input value, which is not the case here.

Key Concepts

Inverse Function

An inverse function is essentially about reversing the roles of inputs and outputs in a given function. When you have a function, say \(f(x)\), its inverse \(f^{-1}(x)\) will take the output \(f(x)\) as input and return \(x\). This means:

• If \(f(a) = b\), then \(f^{-1}(b) = a\).

To have a functional inverse, the original function needs to be one-to-one. A one-to-one function ensures that every output is uniquely tied to one input. This unique pairing is crucial because, if an output is matched to multiple inputs, reversing it would become ambiguous.
Thus, no clear path from output back to input would be possible. This is why checking the one-to-one nature is vital before switching coordinates to form an inverse. In the presence of repetition like in our example with \(g={0,3},{3,7},{6,7},{-2,-2}\), where 7 appears twice, a proper function inverse cannot exist.

Ordered Pairs

Ordered pairs are a fascinating way to express relationships between two entities, often noted as \((x, y)\). Here \(x\) is called the input or domain, while \(y\) is the output or range. When functions are presented as ordered pairs, each pair indicates that a specific input yields a specific output.

• For instance, \((0,3)\) tells us that 0 maps to 3.
• In the case of our function \(g\), the pairs \((3,7)\) and \((6,7)\) indicate that both 3 and 6 map to the same output of 7.

Ordered pairs are perfect for visually examining the behavior of a function. They help in identifying if there are repeated outputs, thus determining the one-to-one nature of a function. Repeated outputs in different pairs point to a many-to-one relationship, ruling out the possibility of a functional inverse.

Output Repetition

Output repetition within a function occurs when different inputs correspond to the same output value, leading to a potential problem when assessing the function's properties. For instance, in the function \(g\) from the exercise, notice the following output repetition:

• The outputs for the inputs 3 and 6 are both 7, showing a duplication.

In a one-to-one function, such output repetition should not occur since each output is expected to be unique to its corresponding input. This uniqueness is fundamental for reversing roles to form an inverse function. When repetition happens, it allows for confusion where a single output could correspond to two or more different inputs, thus losing the needed clarity for an inverse relationship.

Y-Value Analysis

Y-value analysis involves carefully examining the outputs (the \(y\)-values) of each ordered pair in a function to decide if the function is one-to-one. By scanning through the \(y\)-values:

• Identifying repetitions becomes straightforward.
• A function like \(g\) is not one-to-one as we see 7 twice as a \(y\)-value.

Through such an analysis, you can determine whether a function can possibly have an inverse that is also a valid function. If you find no duplicate \(y\)-values, the function is likely one-to-one, meaning each \(x\) maps to a unique \(y\). Therefore, an inverse can be meaningfully created, with every output lined directly with one input, avoiding any mix-up.