Evaluating the Function for Specific Values

Oct 13, 2025 by pynkocean.com 97 views

The function $h$ is defined for all real numbers, and its value depends on the domain of the input $x$ . In this article, we will explore the function $h(x)$ and evaluate its behavior for different domains.

Domain 1: x ≤ -2

For $x \leq -2$ , the function $h(x)$ is defined as $h(x) = 4$ . This means that for any input $x$ less than or equal to -2, the output of the function will be 4.

def h(x):
    if x <= -2:
        return 4

Domain 2: -2 < x < 1

For $-2 < x < 1$ , the function $h(x)$ is defined as $h(x) = (x-1)^2 - 3$ . This is a quadratic function, and its graph is a parabola that opens upwards. The vertex of the parabola is at $x = 1$ , and the minimum value of the function is $-3$ .

def h(x):
    if -2 < x < 1:
        return (x-1)**2 - 3

Domain 3: x ≥ 1

For $x \geq 1$ , the function $h(x)$ is defined as $h(x) = -\frac{1}{4}x + 1$ . This is a linear function, and its graph is a straight line with a negative slope. The y-intercept of the line is at $y = 1$ , and the x-intercept is at $x = 4$ .

def h(x):
    if x >= 1:
        return -0.25*x + 1

Evaluating the Function for Specific Values

Now that we have defined the function $h(x)$ for different domains, let's evaluate its value for some specific inputs.

Evaluating h(-3)

For $x = -3$ , we are in the domain $x \leq -2$ , so the function $h(x)$ is defined as $h(x) = 4$ . Therefore, $h(-3) = 4$ .

Evaluating h(0)

For $x = 0$ , we are in the domain $-2 < x < 1$ , so the function $h(x)$ is defined as $h(x) = (x-1)^2 - 3$ . Therefore, $h(0) = (0-1)^2 - 3 = -4$ .

Evaluating h(2)

For $x = 2$ , we are in the domain $x \geq 1$ , so the function $h(x)$ is defined as $h(x) = -\frac{1}{4}x + 1$ . Therefore, $h(2) = -\frac{1}{4}(2) + 1 = 0.5$ .

Graphing the Function

To visualize the behavior of the function $h(x)$ , we can graph it for different domains.

Graphing h(x) for x ≤ -2

For $x \leq -2$ , the function $h(x)$ is a horizontal line at $y = 4$ . This means that for any input $x$ less than or equal to -2, the output of the function will be 4.

Graphing h(x) for -2 < x < 1

For $-2 < x < 1$ , the function $h(x)$ is a parabola that opens upwards. The vertex of the parabola is at $x = 1$ , and the minimum value of the function is $-3$ .

Graphing h(x) for x ≥ 1

For $x \geq 1$ , the function $h(x)$ is a straight line with a negative slope. The y-intercept of the line is at $y = 1$ , and the x-intercept is at $x = 4$ .

Conclusion

In this article, we have defined the function $h(x)$ for different domains and evaluated its value for specific inputs. We have also graphed the function for different domains to visualize its behavior. The function $h(x)$ is a piecewise function that depends on the domain of the input $x$ . Its value can be 4, a quadratic function, or a linear function, depending on the domain of the input.

References

[1] "Piecewise Functions". Math Open Reference. Retrieved 2023-02-20.
[2] "Quadratic Functions". Math Is Fun. Retrieved 2023-02-20.
[3] "Linear Functions". Math Is Fun. Retrieved 2023-02-20.
Q&A: Evaluating the Function h(x) =====================================

In this article, we will answer some frequently asked questions about the function $h(x)$ and its behavior for different domains.

Q: What is the domain of the function h(x)?

A: The domain of the function $h(x)$ is all real numbers. This means that the function is defined for any input $x$ .

Q: What is the range of the function h(x)?

A: The range of the function $h(x)$ depends on the domain of the input $x$ . For $x \leq -2$ , the range is $[4, \infty)$ . For $-2 < x < 1$ , the range is $(-3, 4)$ . For $x \geq 1$ , the range is $(-\infty, 1)$ .

Q: How do I evaluate the function h(x) for a specific input x?

A: To evaluate the function $h(x)$ for a specific input $x$ , you need to determine which domain the input belongs to. If $x \leq -2$ , then $h(x) = 4$ . If $-2 < x < 1$ , then $h(x) = (x-1)^2 - 3$ . If $x \geq 1$ , then $h(x) = -\frac{1}{4}x + 1$ .

Q: Can I graph the function h(x) for different domains?

A: Yes, you can graph the function $h(x)$ for different domains. For $x \leq -2$ , the graph is a horizontal line at $y = 4$ . For $-2 < x < 1$ , the graph is a parabola that opens upwards. For $x \geq 1$ , the graph is a straight line with a negative slope.

Q: What is the vertex of the parabola for the domain -2 < x < 1?

A: The vertex of the parabola for the domain $-2 < x < 1$ is at $x = 1$ . This is the point where the parabola changes direction.

Q: What is the y-intercept of the line for the domain x ≥ 1?

A: The y-intercept of the line for the domain $x \geq 1$ is at $y = 1$ . This is the point where the line intersects the y-axis.

Q: What is the x-intercept of the line for the domain x ≥ 1?

A: The x-intercept of the line for the domain $x \geq 1$ is at $x = 4$ . This is the point where the line intersects the x-axis.

Q: Can I use the function h(x) in a real-world application?

A: Yes, you can use the function $h(x)$ in a real-world application. For example, you can use it to model the behavior of a physical system that has different behaviors for different input values.

Q: How do I determine which domain the input x belongs to?

A: To determine which domain the input $x$ belongs to, you need to compare the input value to the boundaries of the domains. If $x \leq -2$ , then the input belongs to the domain $x \leq -2$ . If $-2 < x < 1$ , then the input belongs to the domain $-2 < x < 1$ . If $x \geq 1$ , then the input belongs to the domain $x \geq 1$ .

Conclusion

In this article, we have answered some frequently asked questions about the function $h(x)$ and its behavior for different domains. We have also provided examples and explanations to help you understand the function and its applications.