Understanding the Given Points

Oct 13, 2025 by windailys.com 10 views

Introduction

When dealing with functions, understanding the behavior of the graph is crucial. In this article, we will explore the graph of a function $f(x)$ that passes through specific points. The given points are $(0, -2), (1, 0), (-1, -4)$ . Our goal is to determine which of the given functions could be $f(x)$ .

Understanding the Given Points

The given points are $(0, -2), (1, 0), (-1, -4)$ . These points represent the coordinates of the graph of the function $f(x)$ . The x-coordinate represents the input or the value of $x$ , while the y-coordinate represents the output or the value of $f(x)$ .

Analyzing the Points

Let's analyze each point individually:

Point (0, -2): This point indicates that when $x = 0$ , $f(x) = -2$ . This means that the function has a value of -2 at $x = 0$ .
Point (1, 0): This point indicates that when $x = 1$ , $f(x) = 0$ . This means that the function has a value of 0 at $x = 1$ .
Point (-1, -4): This point indicates that when $x = -1$ , $f(x) = -4$ . This means that the function has a value of -4 at $x = -1$ .

Evaluating the Functions

We are given three possible functions: $f(x) = 2x - 2$ , $f(x) = -2x - 2$ , and $f(x) = 2 \sqrt{x} - 2$ . We need to evaluate each function to determine which one could be $f(x)$ .

Function A: $f(x) = 2x - 2$

Let's evaluate this function using the given points:

Point (0, -2): $f(0) = 2(0) - 2 = -2$ . This matches the given point.
Point (1, 0): $f(1) = 2(1) - 2 = 0$ . This matches the given point.
Point (-1, -4): $f(-1) = 2(-1) - 2 = -4$ . This matches the given point.

This function satisfies all the given points, so it could be $f(x)$ .

Function B: $f(x) = -2x - 2$

Let's evaluate this function using the given points:

Point (0, -2): $f(0) = -2(0) - 2 = -2$ . This matches the given point.
Point (1, 0): $f(1) = -2(1) - 2 = -4$ . This does not match the given point.
Point (-1, -4): $f(-1) = -2(-1) - 2 = 0$ . This does not match the given point.

This function does not satisfy all the given points, so it could not be $f(x)$ .

Function C: $f(x) = 2 \sqrt{x} - 2$

Let's evaluate this function using the given points:

Point (0, -2): $f(0) = 2 \sqrt{0} - 2 = -2$ . This matches the given point.
Point (1, 0): $f(1) = 2 \sqrt{1} - 2 = 0$ . This matches the given point.
Point (-1, -4): $f(-1)$ is undefined, as the square root of a negative number is undefined in the real number system. This function does not satisfy all the given points, so it could not be $f(x)$ .

Conclusion

Based on the analysis, the only function that satisfies all the given points is $f(x) = 2x - 2$ . This function has a value of -2 at $x = 0$ , a value of 0 at $x = 1$ , and a value of -4 at $x = -1$ . Therefore, $f(x) = 2x - 2$ could be the function that passes through the given points.

Recommendations

When dealing with functions, it's essential to understand the behavior of the graph. By analyzing the given points, we can determine which function could be $f(x)$ . In this case, the function $f(x) = 2x - 2$ satisfies all the given points, making it a possible candidate for $f(x)$ .

Introduction

In our previous article, we explored the graph of a function $f(x)$ that passes through specific points. We analyzed the given points and evaluated three possible functions to determine which one could be $f(x)$ . In this article, we will answer some frequently asked questions related to the graph of a function.

Q: What is the significance of the given points in determining the graph of a function?

A: The given points are crucial in determining the graph of a function. They represent the coordinates of the graph, where the x-coordinate represents the input or the value of $x$ , and the y-coordinate represents the output or the value of $f(x)$ .

Q: How do we determine which function could be $f(x)$ based on the given points?

A: To determine which function could be $f(x)$ , we need to evaluate each function using the given points. We substitute the x-value of each point into the function and check if the resulting y-value matches the given point. If the function satisfies all the given points, it could be $f(x)$ .

Q: What is the difference between a function and its graph?

A: A function is a mathematical relation between a set of inputs (x-values) and a set of possible outputs (y-values). The graph of a function is a visual representation of the function, where each point on the graph represents a coordinate (x, y) that satisfies the function.

Q: Can a function have multiple graphs?

A: Yes, a function can have multiple graphs. For example, the function $f(x) = x^2$ has two graphs: one in the positive x-axis and one in the negative x-axis.

Q: How do we know if a function is linear or non-linear?

A: A function is linear if it can be represented in the form $f(x) = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. A function is non-linear if it cannot be represented in this form.

Q: Can a function have a negative slope?

A: Yes, a function can have a negative slope. For example, the function $f(x) = -x + 2$ has a negative slope.

Q: How do we determine the domain and range of a function?

A: The domain of a function is the set of all possible x-values that the function can accept. The range of a function is the set of all possible y-values that the function can produce.

Q: Can a function have a domain of all real numbers?

A: Yes, a function can have a domain of all real numbers. For example, the function $f(x) = x^2$ has a domain of all real numbers.

Q: How do we know if a function is one-to-one or many-to-one?

A: A function is one-to-one if each x-value corresponds to a unique y-value. A function is many-to-one if multiple x-values correspond to the same y-value.

Q: Can a function be both one-to-one and many-to-one?

A: No, a function cannot be both one-to-one and many-to-one. A function is either one-to-one or many-to-one, but not both.

Conclusion

In this article, we answered some frequently asked questions related to the graph of a function. We discussed the significance of the given points, how to determine which function could be $f(x)$ , and the difference between a function and its graph. We also covered topics such as linear and non-linear functions, negative slopes, domain and range, and one-to-one and many-to-one functions.