Hey guys! Let's break down how to find the domain and range of the function f(x) = √(cx). It might seem tricky at first, but I promise it’s totally doable once you understand the core concepts. We'll walk through it step by step, so you’ll be a pro in no time! Understanding the domain and range of functions is super important in math because it tells you what inputs you can use and what outputs you can expect. In this case, we're dealing with a square root function, which has its own special rules. So, grab your calculators and let’s dive in!

Understanding the Domain

Okay, let's talk domain! The domain of a function is basically all the possible 'x' values that you can plug into the function without breaking any mathematical rules. For the function f(x) = √(cx), the big rule we need to remember is that you can't take the square root of a negative number (at least, not if you want real number answers). This is the golden rule for square root functions. So, what does this mean for our function? It means that the expression inside the square root, which is 'cx', must be greater than or equal to zero. We can write this as an inequality: cx ≥ 0.

Now, let's consider two cases. If 'c' is a positive number, then 'x' must also be greater than or equal to zero for the inequality to hold true. Think about it: a positive number times a positive number is positive, and a positive number times zero is zero. Both are okay! If 'c' is a negative number, then 'x' must be less than or equal to zero. This is because a negative number times a negative number is positive, and a negative number times zero is still zero. So, the sign of 'c' really determines the possible values of 'x'. If c = 0, then the function becomes f(x) = √(0*x) = 0, which is defined for all real numbers. Therefore, the domain would be all real numbers. To summarize, if c > 0, the domain is x ≥ 0. If c < 0, the domain is x ≤ 0. If c = 0, the domain is all real numbers.

Finding the domain is like setting the rules of the game. It tells you what numbers are allowed to play. Ignoring the domain can lead to all sorts of mathematical chaos, like imaginary numbers where you don't expect them! Therefore, always start by figuring out the domain to ensure your function behaves nicely.

Determining the Range

Alright, now let’s tackle the range! The range of a function is the set of all possible output values (or 'y' values) that the function can produce. For f(x) = √(cx), we need to think about what happens when we take the square root of a number. The square root function always returns a non-negative value. In other words, the result is always zero or positive. This is because the square root is defined as the principal (positive) root. For example, even though both 2 and -2, when squared, give you 4, the square root of 4 is defined as just 2. So, no negative numbers here!

Given that the square root part of the function is always non-negative, the range of f(x) = √(cx) will depend on the constant 'c', but not in the same way as the domain. Since the square root itself is always non-negative, the entire function will always be non-negative as well. Therefore, the range is always y ≥ 0, regardless of whether 'c' is positive or negative. The key here is to focus on what the square root function does. It takes a non-negative number and spits out another non-negative number. This inherent property of the square root dictates that our output will always be zero or positive. It’s like a one-way street: you can only go in the positive direction!

To make it super clear, the range is all the possible 'y' values you get out of the function. Since the square root of any non-negative number is always non-negative, the smallest 'y' value you can get is 0 (when x is such that cx = 0). And because 'x' can be any value within the domain we discussed earlier, the 'y' value can go as high as you want. Therefore, the range is y ≥ 0. Remember, understanding the range is all about understanding what the function does to the 'x' values you plug in.

Examples to Illustrate

Let's nail this down with a few examples! Examples are always helpful to see how this works in practice. Consider f(x) = √(2x). Here, c = 2, which is positive. So, the domain is x ≥ 0. That means we can only plug in zero or positive numbers for 'x'. The range, as we discussed, is y ≥ 0. If we plug in x = 0, we get f(0) = √(20) = 0. If we plug in x = 2, we get f(2) = √(22) = √4 = 2. See how the 'y' values are always zero or positive?

Now, let's try f(x) = √(-3x). In this case, c = -3, which is negative. So, the domain is x ≤ 0. That means we can only plug in zero or negative numbers for 'x'. The range is still y ≥ 0. If we plug in x = 0, we get f(0) = √(-30) = 0. If we plug in x = -3, we get f(-3) = √(-3-3) = √9 = 3. Again, the 'y' values are zero or positive, even though we're plugging in negative 'x' values. One more: if f(x) = √(0x) = 0, the domain is all real numbers and the range is y = 0.

These examples show how the sign of 'c' affects the domain but doesn't change the range. The range is always determined by the square root function itself, which always spits out non-negative values. Practicing with different values of 'c' is a great way to solidify your understanding. Try plugging in different numbers and see how the domain and range play out. Remember, math becomes much easier when you practice!

Graphing and Visualizing

One of the best ways to truly understand the domain and range is to visualize the function using a graph. When you graph f(x) = √(cx), you'll notice some key features. If c > 0, the graph starts at the origin (0,0) and extends to the right, always staying above the x-axis. This visually confirms that the domain is x ≥ 0 and the range is y ≥ 0. The graph will look like a curve that increases as 'x' increases. It starts flat and then gradually rises. Think of it as a plant growing taller as it gets more sunlight.

If c < 0, the graph also starts at the origin (0,0), but it extends to the left, still staying above the x-axis. This visually confirms that the domain is x ≤ 0 and the range is y ≥ 0. The graph will be a mirror image of the previous graph, reflected over the y-axis. It still starts flat and gradually rises, but it does so in the negative 'x' direction. For the special case of f(x) = √(0x) = 0, the graph is a horizontal line along the x-axis, confirming that the domain is all real numbers and the range is y = 0. Seeing these graphs helps connect the algebraic concepts of domain and range to a visual representation. You can use graphing calculators or online tools to plot these functions and see how changing 'c' affects the shape and direction of the graph. Trust me, it's super helpful!

Common Mistakes to Avoid

Even with a solid understanding, it's easy to make mistakes. One common mistake is forgetting that the expression inside the square root must be non-negative. People sometimes blindly plug in numbers without checking if they're allowed, leading to imaginary numbers. Always remember to check the inequality cx ≥ 0 before you start plugging in values.

Another mistake is thinking that the range can be negative. Since the square root function always returns non-negative values, the range will always be y ≥ 0, regardless of the value of 'c'. Some people get confused by the 'c' value and think it affects the range, but it only affects the domain. One more common mistake is messing up the inequality when 'c' is negative. Remember, if 'c' is negative, you need to flip the inequality sign when solving for 'x'. For example, if -2x ≥ 0, then x ≤ 0. Avoiding these common mistakes will help you stay on track and solve these problems accurately.

Conclusion

So, there you have it! Finding the domain and range of f(x) = √(cx) is all about understanding the properties of the square root function and paying attention to the constant 'c'. Remember to always check the inequality cx ≥ 0 to determine the domain, and keep in mind that the range will always be y ≥ 0. By working through examples, graphing the function, and avoiding common mistakes, you'll be a pro in no time. Keep practicing, and you'll master this concept in no time!