Suppose That The Function H H H Is Defined, For All Real Numbers, As Follows:$[ H(x) = \begin{cases} 4 & \text{if } X \leq -2 \ (x-1)^2 - 3 & \text{if } -2 \ \textless \ X \ \textless \ 1 \ -\frac{1}{4}x + 1 & \text{if } X \geq 1

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.