Graphing Sqrt(x) & Sqrt(1-x): Sum, Product, & More

Hey guys! Today, we're diving deep into the world of functions and graphs, specifically looking at how to graph the functions f(x) = √x and g(x) = √(1 - x). But that's not all! We're also going to explore how to graph their sum, product, differences, and quotients. So, buckle up, grab your graphing calculators (or your favorite online graphing tool), and let's get started!

Understanding the Basic Functions

Before we jump into the combined operations, let's make sure we have a solid understanding of our two main functions:

1. f(x) = √x

When dealing with f(x) = √x, the square root function, it’s crucial to understand its fundamental behavior. This function essentially asks, "What number, when multiplied by itself, equals x?" Because we're working with real numbers, we can't take the square root of a negative number (we'd end up in the realm of imaginary numbers, which is a topic for another day!). Therefore, the domain of f(x) is x ≥ 0. This means that our graph will only exist for non-negative x-values.

Graphically, f(x) = √x starts at the origin (0, 0) and increases gradually as x increases. The graph curves upwards, becoming less steep as x gets larger. This is because the rate of change of the square root function decreases as x increases. For example, the difference between √4 (which is 2) and √9 (which is 3) is smaller than the difference between √1 (which is 1) and √4 (which is 2).

To accurately plot f(x) = √x, you can pick a few key points. Let's take a look at some of these points. When x is 0, f(x) is √0, which equals 0. So, we have the point (0, 0). When x is 1, f(x) is √1, which equals 1, giving us the point (1, 1). For x = 4, f(x) is √4, which is 2, leading to the point (4, 2). And when x is 9, f(x) is √9, which equals 3, resulting in the point (9, 3). By plotting these points and connecting them with a smooth curve, we get a clear picture of the square root function's behavior. This curve starts at the origin and gradually rises, illustrating the function's increasing yet decelerating nature. Understanding this behavior is crucial for graphing f(x) accurately and for grasping how it interacts with other functions, like g(x), in subsequent operations.

2. g(x) = √(1 - x)

Now, let’s tackle g(x) = √(1 - x). This function is a slight twist on the square root function we just explored, and it introduces a crucial change inside the square root. The expression inside the square root is now (1 - x), which significantly impacts the domain and the graph’s behavior. To ensure we're taking the square root of a non-negative number, we need to make sure that (1 - x) is greater than or equal to zero. Mathematically, this is expressed as 1 - x ≥ 0. Solving this inequality gives us x ≤ 1.

So, the domain of g(x) is x ≤ 1. This tells us that the graph of g(x) will only exist for x-values that are less than or equal to 1. Unlike f(x), which started at x = 0 and extended to positive x-values, g(x) starts at x = 1 and extends to negative x-values. This is a key difference to keep in mind.

The graph of g(x) = √(1 - x) also has a characteristic shape, but it’s reflected and shifted compared to f(x). To visualize this, consider the function's behavior as x approaches 1. When x is 1, g(x) becomes √(1 - 1) = √0, which is 0. This gives us the point (1, 0) on the graph. Now, as x decreases from 1, the value inside the square root (1 - x) increases, and so does g(x). This means the graph will increase as we move to the left from x = 1.

To plot g(x) accurately, we can again pick a few key points. When x = 1, g(x) = 0, so we have the point (1, 0). If we let x be 0, g(x) is √(1 - 0) = √1, which is 1. This gives us the point (0, 1). For x = -3, g(x) is √(1 - (-3)) = √4, which equals 2, resulting in the point (-3, 2). By plotting these points and smoothly connecting them, we observe that the graph of g(x) starts at (1, 0) and curves upwards as x moves towards negative values. This reflection and shift compared to f(x) are crucial aspects of g(x)'s behavior. Understanding these transformations is essential when we start combining f(x) and g(x) using different mathematical operations.

Combining the Functions

Now for the fun part! We're going to explore what happens when we combine f(x) and g(x) using basic arithmetic operations.

(a) Sum: f(x) + g(x)

The sum of two functions, f(x) + g(x), involves adding their respective y-values for each x-value in their shared domain. This means for each x, we calculate f(x) and g(x) separately and then add the results together. Graphically, this can be visualized as adding the heights of the two graphs at each point along the x-axis. However, there's a crucial consideration we need to make before we dive into the addition process: the domain. The domain of the sum function is not simply the combination of the domains of f(x) and g(x); instead, it's the intersection of their domains. This means we can only add the functions where both f(x) and g(x) are defined.

We already know that the domain of f(x) = √x is x ≥ 0, meaning f(x) exists for all non-negative x-values. On the other hand, the domain of g(x) = √(1 - x) is x ≤ 1, so g(x) is defined for x-values less than or equal to 1. When we find the intersection of these domains, we're looking for the x-values that satisfy both conditions simultaneously. In this case, the shared domain is 0 ≤ x ≤ 1. This is the interval where both f(x) and g(x) are defined, and it’s within this interval that we can meaningfully add the functions.

To graph the sum f(x) + g(x), we start by focusing on the interval 0 ≤ x ≤ 1. Within this interval, we can pick several x-values and calculate the corresponding y-values for both f(x) and g(x), and then add them together. For example, when x = 0, f(0) = √0 = 0, and g(0) = √(1 - 0) = √1 = 1. Adding these together gives us 0 + 1 = 1, so the point (0, 1) is on the graph of the sum function. At x = 1, f(1) = √1 = 1, and g(1) = √(1 - 1) = √0 = 0. The sum at this point is 1 + 0 = 1, so the point (1, 1) is also on the graph. We can also consider the point where x = 0.5. Here, f(0.5) = √0.5 ≈ 0.707, and g(0.5) = √(1 - 0.5) = √0.5 ≈ 0.707. Adding these gives us approximately 1.414, so the point (0.5, 1.414) is part of our sum function’s graph.

By plotting several such points and connecting them smoothly, we can sketch the graph of f(x) + g(x). The graph typically starts at a certain y-value at x = 0, rises to a maximum value within the interval 0 < x < 1, and then comes back down to another y-value at x = 1. The exact shape depends on the specific functions being added, but in the case of f(x) = √x and g(x) = √(1 - x), the graph forms a smooth curve that reflects the combined behavior of both functions within their shared domain. This graphical representation provides a visual understanding of how the sum of two functions behaves, illustrating the combined effect of their individual behaviors.

(b) Product: f(x) * g(x)

Just like with the sum, the product of two functions, f(x) * g(x), involves combining their y-values, but this time we're multiplying them instead of adding. For each x-value in their shared domain, we calculate f(x) and g(x) and then multiply the results together. The shared domain is crucial here, and as we established earlier, the domain where both f(x) = √x and g(x) = √(1 - x) are defined is 0 ≤ x ≤ 1. This interval is the only range where the product f(x) * g(x) will have meaningful real values.

To understand the graph of the product function, we must first revisit the domains of the original functions. As a reminder, f(x) = √x is defined for x ≥ 0, and g(x) = √(1 - x) is defined for x ≤ 1. Thus, their product is only meaningful where both are defined, which is the interval 0 ≤ x ≤ 1. This restricted domain significantly shapes the behavior of the product function.

Now, let’s delve into how we can graph this product. The process starts by evaluating the product f(x) * g(x) = √x * √(1 - x) at several points within the interval 0 ≤ x ≤ 1. By understanding the function's behavior at these key points, we can sketch the graph more accurately. At the endpoints of the interval, the behavior is particularly noteworthy. When x = 0, f(0) = √0 = 0, and g(0) = √(1 - 0) = √1 = 1. Therefore, f(0) * g(0) = 0 * 1 = 0. This gives us the point (0, 0) on the graph. Similarly, when x = 1, f(1) = √1 = 1, and g(1) = √(1 - 1) = √0 = 0. Consequently, f(1) * g(1) = 1 * 0 = 0, resulting in the point (1, 0) on the graph.

These endpoints are crucial because they indicate that the graph of the product function touches the x-axis at both x = 0 and x = 1. Next, we can examine the behavior of the function within the interval. One approach is to choose a midpoint, such as x = 0.5, and evaluate the function there. At x = 0.5, f(0.5) = √0.5, and g(0.5) = √(1 - 0.5) = √0.5. Thus, f(0.5) * g(0.5) = √0.5 * √0.5 = 0.5. This gives us the point (0.5, 0.5) on the graph.

Plotting additional points will further refine our understanding. For example, we could consider x = 0.25 and x = 0.75. At x = 0.25, f(0.25) = √0.25 = 0.5, and g(0.25) = √(1 - 0.25) = √0.75 ≈ 0.866. So, f(0.25) * g(0.25) ≈ 0.5 * 0.866 ≈ 0.433. This results in the point (0.25, 0.433). By symmetry, we might expect the value at x = 0.75 to be similar due to the symmetry of the functions around x = 0.5. Calculating f(0.75) and g(0.75) confirms this symmetry, and we find a corresponding point (0.75, approximately 0.433).

By connecting these plotted points with a smooth curve, we can sketch the graph of f(x) * g(x). The graph starts at the origin (0, 0), rises to a peak somewhere in the middle of the interval (around x = 0.5), and then decreases back to zero at x = 1. This shape reflects the interplay between the two functions: as √x increases and √(1 - x) decreases, their product forms a curved shape that is zero at the boundaries and peaks in the center. The resulting graph provides a visual representation of the product function's behavior, showcasing how the multiplication of two square root functions with complementary arguments produces a curve that encapsulates their interaction.

(c) Differences: f(x) - g(x) and g(x) - f(x)

When we talk about the difference of two functions, it introduces a bit more complexity compared to addition or multiplication because the order of subtraction matters. Specifically, we'll be looking at both f(x) - g(x) and g(x) - f(x). The order changes the sign of the result, and this has a significant impact on the graph.

Let’s first consider f(x) - g(x). This operation involves subtracting the y-value of g(x) from the y-value of f(x) for each x in their shared domain. As before, the shared domain is crucial, and it remains 0 ≤ x ≤ 1, where both functions are defined. Graphically, this means we are looking at the vertical distance between the two graphs at each point, with the distance being positive if f(x) is above g(x) and negative if f(x) is below g(x).

To graph f(x) - g(x), we start by understanding its behavior at the boundaries of the domain. At x = 0, f(0) = √0 = 0, and g(0) = √(1 - 0) = 1. Therefore, f(0) - g(0) = 0 - 1 = -1. This gives us the point (0, -1) on the graph. At x = 1, f(1) = √1 = 1, and g(1) = √(1 - 1) = 0. Thus, f(1) - g(1) = 1 - 0 = 1, resulting in the point (1, 1) on the graph.

To capture the function’s behavior within the interval, we can evaluate it at several intermediate points. At x = 0.5, f(0.5) = √0.5 ≈ 0.707, and g(0.5) = √(1 - 0.5) = √0.5 ≈ 0.707. Hence, f(0.5) - g(0.5) ≈ 0.707 - 0.707 = 0. This gives us the point (0.5, 0) on the graph, indicating a crucial point where the difference is zero, implying that the graphs of f(x) and g(x) intersect at this x-value.

By plotting additional points, we refine our understanding of the graph’s shape. For instance, at x = 0.25, f(0.25) = √0.25 = 0.5, and g(0.25) = √(1 - 0.25) = √0.75 ≈ 0.866. So, f(0.25) - g(0.25) ≈ 0.5 - 0.866 ≈ -0.366, resulting in the point (0.25, -0.366). At x = 0.75, f(0.75) = √0.75 ≈ 0.866, and g(0.75) = √(1 - 0.75) = √0.25 = 0.5. Therefore, f(0.75) - g(0.75) ≈ 0.866 - 0.5 ≈ 0.366, giving us the point (0.75, 0.366).

By smoothly connecting these plotted points, we can sketch the graph of f(x) - g(x). The graph starts at -1 at x = 0, rises to cross the x-axis at x = 0.5, and continues upwards to reach 1 at x = 1. This shape reflects the change in dominance between the two functions: g(x) is larger than f(x) for x < 0.5, leading to negative differences, and f(x) is larger than g(x) for x > 0.5, resulting in positive differences.

Now, let's consider the difference in the opposite order, g(x) - f(x). This operation involves subtracting the y-value of f(x) from the y-value of g(x) for each x in their shared domain. Consequently, g(x) - f(x) is simply the negative of f(x) - g(x), as the subtraction order is reversed. This means that the graph of g(x) - f(x) is a reflection of the graph of f(x) - g(x) across the x-axis. To illustrate, consider the key points we found for f(x) - g(x):

• At x = 0, f(0) - g(0) = -1, so g(0) - f(0) = 1, giving us the point (0, 1).
• At x = 1, f(1) - g(1) = 1, so g(1) - f(1) = -1, resulting in the point (1, -1).
• At x = 0.5, f(0.5) - g(0.5) = 0, so g(0.5) - f(0.5) = 0, yielding the point (0.5, 0), which remains unchanged.

With these points and the understanding that the graph is a reflection, we can sketch the graph of g(x) - f(x). It starts at 1 at x = 0, descends to cross the x-axis at x = 0.5, and continues downwards to reach -1 at x = 1. This graph is a mirror image of the graph of f(x) - g(x), emphasizing the impact of the order of subtraction.

In summary, while the domain remains the same for both f(x) - g(x) and g(x) - f(x), the sign change caused by the order of subtraction results in graphs that are reflections of each other across the x-axis. This comparison highlights how crucial the order of operations is when dealing with function differences, as it flips the perspective from which we view the relationship between the two functions.

(d) Quotients: f(x) / g(x) and g(x) / f(x)

Finally, let’s explore the quotients of our functions, which means we'll be looking at both f(x) / g(x) and g(x) / f(x). Like with differences, the order here is crucial and will significantly impact the resulting graphs. But even more important than the order is the issue of division by zero, which we need to address carefully.

Let's start with f(x) / g(x). This means we're dividing the y-value of f(x) by the y-value of g(x) for each x-value. Remember, f(x) = √x and g(x) = √(1 - x). The first thing we need to consider is the domain. We know the shared domain where both functions are defined is 0 ≤ x ≤ 1. However, we have an additional restriction now: we cannot divide by zero. So, we need to find where g(x) = 0, because that's where f(x) / g(x) will be undefined.

g(x) = √(1 - x) is equal to zero when 1 - x = 0, which means x = 1. So, x = 1 is not in the domain of f(x) / g(x). Our domain is now 0 ≤ x < 1. We exclude x = 1 because it would lead to division by zero.

To graph f(x) / g(x), we need to evaluate the function f(x) / g(x) = √x / √(1 - x) over the domain 0 ≤ x < 1. Let's start by looking at the endpoints of our interval. At x = 0, f(0) = √0 = 0, and g(0) = √(1 - 0) = 1. Therefore, f(0) / g(0) = 0 / 1 = 0. So, the point (0, 0) is on the graph.

However, as x approaches 1 from the left, we have an interesting situation. The numerator, √x, approaches √1 = 1, but the denominator, √(1 - x), approaches 0. This means that the quotient f(x) / g(x) approaches infinity as x approaches 1. This indicates that we have a vertical asymptote at x = 1.

Now, let's consider some intermediate points. At x = 0.25, f(0.25) = √0.25 = 0.5, and g(0.25) = √(1 - 0.25) = √0.75 ≈ 0.866. Thus, f(0.25) / g(0.25) ≈ 0.5 / 0.866 ≈ 0.577, giving us the point (0.25, 0.577). At x = 0.5, f(0.5) = √0.5 ≈ 0.707, and g(0.5) = √(1 - 0.5) = √0.5 ≈ 0.707. So, f(0.5) / g(0.5) ≈ 0.707 / 0.707 = 1, resulting in the point (0.5, 1). At x = 0.75, f(0.75) = √0.75 ≈ 0.866, and g(0.75) = √(1 - 0.75) = √0.25 = 0.5. Therefore, f(0.75) / g(0.75) ≈ 0.866 / 0.5 ≈ 1.732, giving us the point (0.75, 1.732).

By plotting these points and considering the vertical asymptote, we can sketch the graph of f(x) / g(x). The graph starts at (0, 0), increases as x increases, and shoots off to infinity as x approaches 1. This behavior is characteristic of functions with vertical asymptotes.

Now, let’s look at the reciprocal quotient, g(x) / f(x). This time, we're dividing the y-value of g(x) by the y-value of f(x). So, we have g(x) / f(x) = √(1 - x) / √x. Again, we need to consider the domain. We still have the shared domain 0 ≤ x ≤ 1, but we now need to exclude points where f(x) = 0, because that would mean dividing by zero. f(x) = √x is equal to zero when x = 0. So, x = 0 is not in the domain of g(x) / f(x). Our domain is now 0 < x ≤ 1.

To graph g(x) / f(x), we follow a similar process as before. First, we consider the endpoints. As x approaches 0 from the right, the numerator, √(1 - x), approaches √(1 - 0) = 1, but the denominator, √x, approaches 0. This means that the quotient g(x) / f(x) approaches infinity as x approaches 0. So, we have a vertical asymptote at x = 0.

At x = 1, g(1) = √(1 - 1) = 0, and f(1) = √1 = 1. Therefore, g(1) / f(1) = 0 / 1 = 0. So, the point (1, 0) is on the graph.

Now, let's consider the same intermediate points as before, but this time we'll calculate g(x) / f(x). At x = 0.25, g(0.25) = √0.75 ≈ 0.866, and f(0.25) = √0.25 = 0.5. Thus, g(0.25) / f(0.25) ≈ 0.866 / 0.5 ≈ 1.732, giving us the point (0.25, 1.732). At x = 0.5, g(0.5) = √0.5 ≈ 0.707, and f(0.5) = √0.5 ≈ 0.707. So, g(0.5) / f(0.5) ≈ 0.707 / 0.707 = 1, resulting in the point (0.5, 1). At x = 0.75, g(0.75) = √0.25 = 0.5, and f(0.75) = √0.75 ≈ 0.866. Therefore, g(0.75) / f(0.75) ≈ 0.5 / 0.866 ≈ 0.577, giving us the point (0.75, 0.577).

By plotting these points and considering the vertical asymptote, we can sketch the graph of g(x) / f(x). The graph starts at infinity as x approaches 0, decreases as x increases, and reaches 0 at x = 1. This behavior is the inverse of what we saw for f(x) / g(x).

In summary, the quotients f(x) / g(x) and g(x) / f(x) demonstrate the importance of considering the domain and the potential for division by zero. They also show how swapping the numerator and denominator can significantly change the behavior of the function, leading to different asymptotes and graph shapes.

Conclusion

Wow, we covered a lot! We successfully graphed f(x) = √x and g(x) = √(1 - x), and then we dove into graphing their sums, products, differences, and quotients. Remember, guys, the key takeaways are understanding the individual functions, identifying their domains, and carefully considering how operations like addition, multiplication, subtraction, and division affect the combined functions. Keep practicing, and you'll become a graphing pro in no time! Thanks for joining me on this mathematical adventure!