What Is The Domain And Range Of $y = -2 \sqrt{x-3} + 4$?
Introduction
When dealing with functions, it's essential to understand the domain and range of a function. The domain of a function is the set of all possible input values (x-values) for which the function is defined, while the range is the set of all possible output values (y-values) that the function can produce. In this article, we will explore the domain and range of the given function $y = -2 \sqrt{x-3} + 4$.
Understanding the Function
The given function is a quadratic function with a square root term. The square root term is $\sqrt{x-3}$, which means that the value inside the square root must be non-negative. This implies that $x-3 \geq 0$, or $x \geq 3$. This is the first constraint on the domain of the function.
Domain of the Function
The domain of the function is the set of all possible input values (x-values) for which the function is defined. In this case, the function is defined for all values of x that satisfy the inequality $x \geq 3$. This means that the domain of the function is $[3, \infty)$.
Range of the Function
The range of the function is the set of all possible output values (y-values) that the function can produce. To find the range, we need to consider the behavior of the function as x approaches positive and negative infinity.
Finding the Range
As x approaches positive infinity, the value of $\sqrt{x-3}$ approaches infinity, and the value of $-2 \sqrt{x-3}$ approaches negative infinity. As x approaches negative infinity, the value of $\sqrt{x-3}$ approaches negative infinity, and the value of $-2 \sqrt{x-3}$ approaches positive infinity.
Determining the Range
Since the function is a quadratic function with a square root term, the range is determined by the minimum and maximum values of the function. To find the minimum and maximum values, we need to find the vertex of the function.
Finding the Vertex
The vertex of the function is the point where the function changes from decreasing to increasing or vice versa. To find the vertex, we need to find the value of x that minimizes the function.
Calculating the Vertex
To find the vertex, we can use the formula $x = -\frac{b}{2a}$, where a and b are the coefficients of the quadratic term and the linear term, respectively. In this case, a = -2 and b = 0, so the vertex is at x = 0.
Finding the Minimum Value
To find the minimum value of the function, we need to substitute the value of x into the function. Substituting x = 0 into the function, we get:
Determining the Range
Since the function is a quadratic function with a square root term, the range is determined by the minimum and maximum values of the function. The minimum value of the function is $-2i\sqrt{3} + 4$, and the maximum value is $\infty$.
Conclusion
In conclusion, the domain of the function $y = -2 \sqrt{x-3} + 4$ is $[3, \infty)$, and the range is $(-\infty, -2i\sqrt{3} + 4]$.
Final Answer
The final answer is:
- Domain: $[3, \infty)$
- Range: $(-\infty, -2i\sqrt{3} + 4]$
Introduction
In our previous article, we explored the domain and range of the function $y = -2 \sqrt{x-3} + 4$. In this article, we will answer some frequently asked questions related to the domain and range of this function.
Q1: What is the domain of the function $y = -2 \sqrt{x-3} + 4$?
A1: The domain of the function is the set of all possible input values (x-values) for which the function is defined. In this case, the function is defined for all values of x that satisfy the inequality $x \geq 3$. This means that the domain of the function is $[3, \infty)$.
Q2: Why is the domain of the function $[3, \infty)$?
A2: The domain of the function is $[3, \infty)$ because the value inside the square root must be non-negative. This implies that $x-3 \geq 0$, or $x \geq 3$.
Q3: What is the range of the function $y = -2 \sqrt{x-3} + 4$?
A3: The range of the function is the set of all possible output values (y-values) that the function can produce. In this case, the range is $(-\infty, -2i\sqrt{3} + 4]$.
Q4: Why is the range of the function $(-\infty, -2i\sqrt{3} + 4]$?
A4: The range of the function is $(-\infty, -2i\sqrt{3} + 4]$ because the function is a quadratic function with a square root term. The minimum value of the function is $-2i\sqrt{3} + 4$, and the maximum value is $\infty$.
Q5: How do I find the domain and range of a function?
A5: To find the domain and range of a function, you need to consider the following:
- For the domain, you need to find the values of x that satisfy the inequality inside the square root.
- For the range, you need to find the minimum and maximum values of the function.
Q6: What is the difference between the domain and range of a function?
A6: The domain of a function is the set of all possible input values (x-values) for which the function is defined, while the range is the set of all possible output values (y-values) that the function can produce.
Q7: Can the domain and range of a function be the same?
A7: Yes, the domain and range of a function can be the same. For example, if the function is a linear function, the domain and range are both the set of all real numbers.
Q8: How do I graph the domain and range of a function?
A8: To graph the domain and range of a function, you can use the following steps:
- Graph the function on a coordinate plane.
- Identify the domain and range of the function.
- Graph the domain and range on the coordinate plane.
Q9: What is the importance of understanding the domain and range of a function?
A9: Understanding the domain and range of a function is important because it helps you to:
- Determine the values of x that satisfy the inequality inside the square root.
- Find the minimum and maximum values of the function.
- Graph the function on a coordinate plane.
Q10: Can the domain and range of a function be negative?
A10: Yes, the domain and range of a function can be negative. For example, if the function is a quadratic function with a negative coefficient, the domain and range can be negative.
Conclusion
In conclusion, understanding the domain and range of a function is essential in mathematics. By following the steps outlined in this article, you can find the domain and range of a function and graph it on a coordinate plane.
Final Answer
The final answer is:
- Domain: $[3, \infty)$
- Range: $(-\infty, -2i\sqrt{3} + 4]$