Problem 22 - Precalculus : Building Concepts and Connections

Question

State whether each function given by a table is one-to-one. Explain your reasoning. $$\begin{aligned}&x \quad f(x)\\\&\begin{array}{cc}-2 & -9 \\\\-1 & -8\\\0&-7\\\1&-6\\\2&-5\end{array}\end{aligned}$$

Step-by-Step Solution

Answer

Yes, the given function is one-to-one. This is because for each x-value, there exists a unique f(x)-value according to the table, which meets the criteria for one-to-one functions.

Step 1: Understand the table

Firstly, we should understand the table that is given. The first row of the table represents the 'x' values or the input of the function. The second row of the table represents the 'f(x)' values or the output of the function. For any given x-value, there corresponds an f(x)-value as per the table. For a function to be one-to-one, every x-value should have a unique f(x)-value; the same f(x)-value should not correspond to more than one x-value.

Step 2: Examination of the table

Examine the table to see if there are any repeated f(x) values for different x-values. The f(x) values are -9, -8, -7, -6, and -5. None of these values are repeated for different x-values, hence, this implies that for each x-value, there is a unique f(x) value.

Step 3: Conclude the result

Since we have verified that each x-value is associated with a unique f(x)-value, we can safely conclude that, according to the definition of one-to-one functions, the given function is indeed one-to-one.

Key Concepts

Understanding the Function Table

A function table is a simple tool that helps us see the relationship between inputs and outputs in a function. Imagine you have two rows like in a grid. The first row lists the 'x' values, which are the inputs. The second row lists the 'f(x)' values, which are the outputs.

This organized way lets us quickly analyze how inputs are paired with outputs. When you look at a function table, remember that each input should be paired with one output only. This unique pairing is crucial to identifying a valid function. In our specific exercise, the table shows inputs from -2 to 2 and their outputs. We observe them arranged straight down in a straightforward manner, making it easy to identify any patterns or repetitions.

By understanding the function table well, you can decide if a function behaves as expected, such as meeting the one-to-one criteria needed for this exercise.

Recognizing Unique Function Values

One major step in determining if a function is one-to-one is checking for unique function values. When studying a function's outputs, the key is that no output should repeat for different inputs. If we find that each input has a distinct output, it confirms the uniqueness required for the function to be one-to-one.

In our table, the outputs are -9, -8, -7, -6, and -5. Each of these values appears only once, even though there are multiple inputs. This pattern shows that each input corresponds to a unique output. Thus, this particular function's outputs are unique, meaning that for each different 'x,' there isn't a shared 'f(x)'.

Recognizing this quality in function values is an essential step for confirming the function's one-to-one nature. This method helps ensure no two different inputs result in the same output.

Applying Precalculus Concepts

Precalculus introduces fundamental ideas like functions and their properties, such as one-to-one nature. A one-to-one function ensures every input maps to a distinct output. This effectively implies no two different inputs can produce the same output. This property is crucial for many applications in higher mathematics and ensures functions can be inverted.

In this exercise, we used precalculus concepts to analyze the given function table. Understanding these properties helps in identifying and analyzing functions reliably.

• Firstly, clarify what a function and its components are, like input and output.
• Next, recognize that one-to-one corresponds to unique outputs for every input.
• Finally, observing patterns in data tables can aid in confirming a function's behavior.

By combining this knowledge, precalculus equips us to discern and delve into more complex mathematical ideas seamlessly. It's a bridge to advanced mathematics, enabling a deeper grasp on concepts like inverse functions and calculus. Through these skills, concluding that the function provided is one-to-one becomes straightforward.