Problem 11 - Beginning and Intermediate Algebra

Question

Graph each function by plotting points, and identify the domain and range. $$f(x)=2 \sqrt{x}$$

Step-by-Step Solution

Answer

The function \(f(x)=2 \sqrt{x}\) can be graphed by plotting points. Some example points include (0,0), (1,2), (4,4), and (9,6). The domain of the function is \(\{x \in \mathbb{R} | x \geq 0\}\) and the range is \(\{f(x) \in \mathbb{R} | f(x) \geq 0\}\).

Step 1: Graph the function by plotting points

First, we will plot several points to help us graph the function. The function is given by \(f(x)=2 \sqrt{x}\). Choose some values of x, and calculate the corresponding values of f(x). Here are some example points: - x = 0: \(f(x)=2 \sqrt{0}=0\) - x = 1: \(f(x)=2 \sqrt{1}=2\) - x = 4: \(f(x)=2 \sqrt{4}=4\) - x = 9: \(f(x)=2 \sqrt{9}=6\) After calculating the values of f(x), plot these points on a graph and connect them to create a smooth curve.

Step 2: Identify the domain of the function

The domain is the set of all possible values of x for which the function is defined. Since the function involves a square root, it will only be defined for non-negative values of x (x ≥ 0). Therefore, the domain of the function can be written as: \[Domain = \{x \in \mathbb{R} | x \geq 0\}\]

Step 3: Identify the range of the function

The range is the set of all possible values of f(x) for the given function. Since the function can only output non-negative values (because the square root of a non-negative number is always non-negative), the range of the function is: \[Range = \{f(x) \in \mathbb{R} | f(x) \geq 0\}\]

Step 4: Conclusion

In summary, the function \(f(x)=2 \sqrt{x}\) can be graphed by plotting points and has a domain of \(\{x \in \mathbb{R} | x \geq 0\}\) and a range of \(\{f(x) \in \mathbb{R} | f(x) \geq 0\}\).

Key Concepts

Domain and Range

When we talk about the domain of a function, we refer to all the possible input values, or \(x\)-values, that you can use in the function without encountering mathematical issues. For the square root function \(f(x)=2\sqrt{x}\), we need to remember that we can't take the square root of a negative number without introducing complex numbers, which we aren't dealing with in this context. Therefore, the domain for this function is all non-negative real numbers, or \(x \geq 0\). This means that you can plug in zero and any positive number into the function, as \(f(x)\) will compute normally in those cases.

The range, on the other hand, refers to all the possible output values, or \(f(x)\)-values, that the function can produce from its domain. In the case of \(f(x)=2\sqrt{x}\), since the square root function cannot produce negative results and multiplying by 2 only stretches the output, every value of \(f(x)\) is also non-negative. Therefore, the range is \(f(x) \geq 0\). This shows us that the smallest value of \(f(x)\) is zero, and there is no upper limit because increasing \(x\) can make \(f(x)\) as large as we like.

Square Root Function

The square root function \(\sqrt{x}\) is fundamental in mathematics and appears often in functions, like in \(f(x)=2\sqrt{x}\). You might wonder what this function does in terms of graphing. The square root function takes a value \(x\) and returns another number whose square is \(x\). For instance, \(\sqrt{4} = 2\) because \(2^2 = 4\).

In terms of graphing, the square root function generally starts at the origin \((0,0)\) and increases gradually as \(x\) increases, creating a curve that rises to the right. This is because small increases in \(x\) early on yield smaller increases in \(f(x)\) compared to when \(x\) is larger. By multiplying by 2 in \(f(x)=2\sqrt{x}\), each \(f(x)\) value is doubled, resulting in a graph that looks stretched vertically compared to the basic square root function. Overall, this function smoothly increases and never decreases.

Plotting Points

Before plotting a function, you need some sample points that tell you where to position the curve on the graph. With \(f(x)=2\sqrt{x}\), select a few values of \(x\) that are convenient to compute. Start with \(x=0\), because it often gives a clear starting point for many functions. Here, \(f(0)=0\), so you plot at the origin (0, 0).

Next, take \(x=1\). The function \(f(x)=2\sqrt{1}=2\) indicates you plot the point (1, 2). These points help clarify the pattern of the curve. Consider additional points like \(x=4\) where \(f(x)=4\), and \(x=9\) where \(f(x)=6\). Plot these points to outline the curve, and gently connect them with a smooth curve. This results in a graph which accurately represents the function's behavior. Remember, accuracy in plotting is key to reflecting the true nature of a function.