Problem 84
Question
For the following exercises, graph \(y=\sqrt{x}\) on the given viewing window. Determine the corresponding range for each viewing window. Show each graph. $$ [0,10,000] $$
Step-by-Step Solution
Verified Answer
Range for \(y=\sqrt{x}\) from \([0, 10000]\) is \([0, 100]\). Graph shows a curve from (0,0) to (10000, 100).
1Step 1: Understanding the Function
The equation given is \( y = \sqrt{x} \). This is the square root function, which means for each value of \( x \), \( y \) will be the square root of \( x \). This function only takes non-negative values of \( x \), which correlates with the viewing window \([0, 10000]\).
2Step 2: Applying the Domain to Find Range
Given the viewing window \([0, 10000]\), the domain of the function is exactly this interval. The domain is all possible \( x \) values, meaning \( x \) ranges from 0 to 10,000. Hence, the range will be the square root of these values, \( \sqrt{x} \).
3Step 3: Calculating Range Limits
Calculate the smallest and largest values of \( \sqrt{x} \) within the domain. For \( x = 0 \), \( y = \sqrt{0} = 0 \). For \( x = 10000 \), \( y = \sqrt{10000} = 100 \). Thus, the range is \([0, 100]\).
4Step 4: Graphing the Function
Graph the function \( y = \sqrt{x} \) on the given viewing window \([0, 10000]\). The graph is a curve that starts at the origin (0,0) and rises steadily to the point (10000, 100). It is important to show this gradual increase as \( x \) increases on the graph.
Key Concepts
Domain and RangeSquare Root FunctionFunction Graphing
Domain and Range
In mathematics, every function has a domain and a range. The domain refers to all the possible input values (or x-values) that the function can accept. The range, on the other hand, is all possible output values (or y-values) that result from plugging the domain values into the function.
The square root function, written as \(y = \sqrt{x}\), only allows non-negative numbers for its input. This is because you cannot find the square root of a negative number in the set of real numbers without venturing into complex numbers, which are outside the scope of this discussion.
For our function within the viewing window \([0, 10000]\), the domain is exactly this interval. These are the possible x-values, starting from 0 and stretching all the way to 10,000. Correspondingly, the range will be effected by these x-values.
The square root function, written as \(y = \sqrt{x}\), only allows non-negative numbers for its input. This is because you cannot find the square root of a negative number in the set of real numbers without venturing into complex numbers, which are outside the scope of this discussion.
For our function within the viewing window \([0, 10000]\), the domain is exactly this interval. These are the possible x-values, starting from 0 and stretching all the way to 10,000. Correspondingly, the range will be effected by these x-values.
- Minimum value of \( y \) is \( \sqrt{0} = 0 \)
- Maximum value of \( y \) is \( \sqrt{10000} = 100 \)
Square Root Function
The square root function, \(y = \sqrt{x}\), is a fundamental mathematical function that serves as a building block for many more complex operations. It is particularly interesting due to its unique properties:
The characteristic of the square root function is that it grows slower as x increases. The initial growth is faster when x is closer to zero, and as x continues to rise, this growth diminishes gradually. By inputting values into the function, you can observe how the relationship is non-linear.
- Non-negative domain: it only operates on non-negative numbers
- Smooth curve: it creates a gentle upward curve when plotted
- Non-linear: the rate at which y increases slows down as x gets larger
The characteristic of the square root function is that it grows slower as x increases. The initial growth is faster when x is closer to zero, and as x continues to rise, this growth diminishes gradually. By inputting values into the function, you can observe how the relationship is non-linear.
Function Graphing
Graphing the square root function, \(y = \sqrt{x}\), gives us a visual representation of its behavior. Visualizing functions is not only helpful but also enhances understanding by depicting how input and output values relate on a coordinate system.
When plotting \(y = \sqrt{x}\) on the interval \([0, 10000]\), start with the x-axis ranging from 0 to 10,000. The y-axis, which represents the range, should accommodate values from 0 up to 100. As you graph, you’ll notice the curve starts at the point \((0,0)\) and moves upwards towards \((10000, 100)\).
When plotting \(y = \sqrt{x}\) on the interval \([0, 10000]\), start with the x-axis ranging from 0 to 10,000. The y-axis, which represents the range, should accommodate values from 0 up to 100. As you graph, you’ll notice the curve starts at the point \((0,0)\) and moves upwards towards \((10000, 100)\).
- The graph is a curve, starting slowly and rising more gently as x increases.
- Unlike a straight line, this curve illustrates non-linear growth: rapid at lower x-values but slowing as x gets larger.
Other exercises in this chapter
Problem 83
For the following exercises, use \(f(x)=x^{3}+1\) and \(g(x)=\sqrt[3]{x-1}\). What is the domain of \((f \circ g)(x) ?\)
View solution Problem 84
For the following exercises, use \(f(x)=x^{3}+1\) and \(g(x)=\sqrt[3]{x-1}\). Let \(f(x)=\frac{1}{x}\). a. Find \((f \circ f)(x)\). b. Is \((f \circ f)(x)\) for
View solution Problem 85
Let \(F(x)=(x+1)^{5}, f(x)=x^{5},\) and \(g(x)=x+1\). True or False: \((g \circ f)(x)=F(x)\).
View solution Problem 85
For the following exercises, let \(F(x)=(x+1)^{5}, f(x)=x^{5},\) and \(g(x)=x+1\). True or False: \((g \circ f)(x)=F(x)\).
View solution