Domain & Range Of W(x) = -(3x)^(1/2) - 4 Explained!

Let's break down the function $w(x) = -(3x)^{\frac{1}{2}} - 4$ and figure out its domain and range. Understanding these concepts is fundamental in mathematics, as they define the set of possible input and output values for a function. We'll go step-by-step to make sure everything is crystal clear.

Understanding the Function

First, it's crucial to recognize the function we're dealing with. The function $w(x)$ involves a square root, a transformation, and a vertical shift. The core component is the square root function, $x^{\frac{1}{2}}$ or $\sqrt{x}$, which has inherent restrictions on its domain because we can't take the square root of a negative number (at least not within the realm of real numbers). The '3x' inside the square root means we're scaling the input. The negative sign in front reflects the function over the x-axis, and the '-4' shifts the entire function downward by 4 units. Each of these operations affects the domain and range in specific ways.

The domain represents all possible x-values that you can plug into the function without causing any mathematical errors (like dividing by zero or taking the square root of a negative number). The range represents all possible y-values (or $w(x)$ values in this case) that the function can output. Determining these sets requires a careful examination of the function's components and their transformations. We must consider the restrictions imposed by the square root, the reflection caused by the negative sign, and the vertical shift that alters the function's position on the coordinate plane. Each element contributes to shaping the overall behavior and, consequently, the domain and range of the function. A strong grasp of these principles not only allows us to solve problems but also enriches our comprehension of mathematical connections.

Determining the Domain

The domain of $w(x)$ is restricted by the square root. For the function to be defined in real numbers, the expression inside the square root, $3x$, must be greater than or equal to zero:

$3x \ge 0$

Dividing both sides by 3, we get:

$x \ge 0$

So, the domain of $w(x)$ is all real numbers greater than or equal to 0. In interval notation, this is:

$[0, \infty)$

The domain focuses on what values of x we can actually use in the equation. Think about it – we can't put a negative number under a square root (and get a real number back). So, whatever is inside that square root, in this case 3x, has to be zero or positive. Solving the inequality $3x \ge 0$ gives us $x \ge 0$. This means we can only use zero and positive numbers for x. This limitation is crucial because it dictates the left boundary of our function's existence on the x-axis. If we were to graph this, we wouldn't see anything to the left of x=0. The multiplier '3' inside the square root compresses the graph horizontally, but it doesn't change the fundamental restriction that x must be non-negative. Understanding the domain is pivotal in analyzing and interpreting functions accurately. It provides the framework for predicting the behavior and limitations of the function, ensuring that we're only working with valid and meaningful inputs. This groundwork is essential for further analysis and problem-solving within the realm of mathematics.

Determining the Range

Now let's figure out the range. The basic square root function, $\sqrt{x}$, has a range of $[0, \infty)$. However, $w(x)$ has two transformations that affect the range:

1. The negative sign: The negative sign in front of the square root reflects the function over the x-axis. This means the range becomes $(-\infty, 0]$.
2. The '-4': Subtracting 4 shifts the entire function down by 4 units. This shifts the range to $(-\infty, -4]$.

Therefore, the range of $w(x)$ is $(-\infty, -4]$.

The range, on the other hand, describes all the possible y values that our function can produce. Start with the basic $\sqrt{x}$, which only gives us zero and positive numbers. Now, our function has a negative sign in front, $-(3x)^{\frac{1}{2}}$. This flips the square root function upside down, so now we only get zero and negative numbers. So far, our range is $(-\infty, 0]$. But there's one more thing: we're subtracting 4 from the whole thing. This shifts the entire graph down by 4 units, so our range also shifts down by 4. Therefore, our final range is $(-\infty, -4]$. Understanding how transformations affect the range is essential for visualizing the graph of the function. The negative sign reflects the function, inverting the y-values. The subtraction shifts the entire function down, directly changing the minimum and maximum possible outputs. By analyzing these transformations, we can accurately predict the behavior of the function and its potential outputs, making it easier to interpret and apply in various contexts. This comprehension is key to mastering function analysis and problem-solving.

In Summary

• The domain of $w(x) = -(3x)^{\frac{1}{2}} - 4$ is $[0, \infty)$.
• The range of $w(x) = -(3x)^{\frac{1}{2}} - 4$ is $(-\infty, -4]$.

Understanding domain and range is crucial for working with functions. It tells us the permissible inputs and the possible outputs, allowing us to accurately analyze and interpret mathematical relationships. We explored how transformations affect these characteristics and provided a step-by-step approach to solving similar problems. By understanding these concepts, you'll be better equipped to handle a wide range of mathematical challenges. Domain and range are not just abstract concepts; they provide a framework for analyzing the behavior of functions and making predictions about their outputs. When faced with a new function, systematically examining its components and transformations will help you determine its domain and range accurately. This approach not only enhances your problem-solving skills but also deepens your understanding of fundamental mathematical principles. Keep practicing, and you'll find that analyzing domains and ranges becomes second nature, empowering you to tackle more complex mathematical problems with confidence.

For more information on domain and range, you can check out Khan Academy's article on domain and range.