Finding X For F(x) = -3 Using A Function Table

In mathematics, a function is a fundamental concept that describes a relationship between an input and an output. The function $f(x)$ is a way to represent this relationship, where $x$ is the input and $f(x)$ is the output. In simpler terms, you can think of a function as a machine: you put something in ($x$), the machine does something to it, and you get something out ($f(x)$).

In the given table, we have a set of specific inputs ($x$ values) and their corresponding outputs ($f(x)$ values). This table essentially maps each $x$ to its unique $f(x)$. Understanding how to read and interpret this table is crucial for solving problems related to functions. This involves recognizing that each row represents a pair of input and output values, and by looking at the table, we can determine the output for a given input, or vice versa.

For instance, the table tells us that when $x$ is -4, the function $f(x)$ gives us -66. Mathematically, we write this as $f(-4) = -66$. Similarly, when $x$ is 0, $f(0) = -2$. These individual points are essentially snapshots of the function's behavior. Each pair $(x, f(x))$ can be plotted as a point on a graph, and collectively, these points can give us a visual representation of the function. By analyzing these points, we can start to infer the nature of the function – whether it's linear, quadratic, or something else. However, with a limited number of points, as in this table, it's challenging to definitively determine the function's exact form. Instead, we focus on answering specific questions based on the provided data. The essence of working with functions lies in the ability to connect inputs and outputs and use this relationship to solve problems and make predictions. This foundational concept is essential in various fields, from mathematics and physics to computer science and economics.

The table presented in the problem is a concise way of representing a function's behavior at specific points. Let's delve deeper into how to interpret this table effectively. The table has two columns: the first column lists the input values, denoted by $x$, and the second column lists the corresponding output values, denoted by $f(x)$. Each row in the table represents a pair of values – an input and its corresponding output. This pairing is the core concept of a function: for each input, there is exactly one output.

To read the table, you simply look across a row to find the corresponding values. For example, if you look at the row where $x = -3$, you'll see that $f(x) = -29$. This means that when the input to the function $f$ is -3, the output is -29. Mathematically, we express this as $f(-3) = -29$. This notation is crucial for understanding and communicating mathematical concepts effectively. Similarly, if you look at the row where $x = 2$, you find that $f(2) = 6$. This tells us that when the input is 2, the function's output is 6.

Understanding how to extract information from such a table is vital for solving problems involving functions. The table provides us with specific data points, which we can use to answer questions, make predictions, or even deduce the nature of the function itself. While the table doesn't give us the function's equation directly, it provides concrete examples of the function's behavior. These examples are powerful tools for understanding and working with functions. For instance, we can use these points to estimate the function's behavior between the given values, or to identify patterns and trends. However, it's important to remember that the table represents the function only at these specific points; the function could behave differently at other values of $x$.

The core question we need to address is: when $f(x) = -3$, what is the value of $x$? This question is asking us to reverse the usual process of evaluating a function. Instead of being given an input ($x$) and finding the output ($f(x)$), we are given the output and need to find the corresponding input.

To solve this, we need to carefully examine the table. We are looking for the row where the $f(x)$ value is equal to -3. Scanning the second column, we can identify the row where $f(x)$ is indeed -3. Once we've found this row, we simply read across to the first column to find the corresponding $x$ value. This value of $x$ is the solution to our problem.

In this specific table, we can see that when $x = -1$, the value of $f(x)$ is -3. Therefore, the solution to the question “When $f(x) = -3$, what is $x$?” is $x = -1$. This process highlights the importance of understanding the relationship between inputs and outputs in a function. By carefully analyzing the table, we can effectively reverse the function's operation and find the input that produces a given output. This skill is fundamental in various mathematical contexts, including solving equations, graphing functions, and understanding mathematical models.

Now, let's analyze the provided options to ensure we select the correct answer. We have the following options:

• A. -29
• B. -10
• C. -3
• D. -1

We've already determined that the correct answer is $x = -1$. Let's see why the other options are incorrect. Option A, -29, is the value of $f(x)$ when $x = -3$, not the other way around. Option B, -10, is the value of $f(x)$ when $x = -2$. Option C, -3, is the value of $f(x)$, not the value of $x$. It's crucial to differentiate between the input ($x$) and the output ($f(x)$) when solving these types of problems. The table provides pairs of $(x, f(x))$ values, and we must correctly identify which value corresponds to the given condition.

Therefore, by carefully analyzing the options and comparing them with the information in the table, we can confidently confirm that the correct answer is D. -1. This methodical approach of eliminating incorrect options is a valuable strategy for problem-solving in mathematics and other fields. It not only helps in arriving at the correct answer but also reinforces the understanding of the underlying concepts.

In conclusion, by carefully analyzing the table representing the function $f(x)$, we can determine the value of $x$ when $f(x) = -3$. The correct answer is D. -1. This exercise highlights the importance of understanding the relationship between inputs and outputs in a function and the ability to extract information from a table representing a function's values. This skill is crucial for solving problems related to functions and for understanding mathematical concepts in general. Remember, the key is to carefully read the table, identify the given information, and use it to answer the question accurately.

This problem provides a foundation for more complex concepts in mathematics, such as finding the inverse of a function or analyzing the graph of a function. By mastering these basic skills, you'll be well-equipped to tackle more challenging problems in the future. The ability to interpret and utilize tabular data is not only essential in mathematics but also in various fields that rely on data analysis and interpretation.