Problem 86

Question

For the following exercises, graph $y=\sqrt[3]{x}$ on the given viewing window. Determine the corresponding range for each viewing window. Show each graph. $$ [-1,000,1,000] $$

Step-by-Step Solution

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Answer

The range for the viewing window is all real numbers.

1Step 1: Understand the function

The function given is $ y = \sqrt[3]{x} $, which represents the cube root of $ x $. The cube root function is defined for all real numbers and can have both positive and negative values.

2Step 2: Determine the domain

For the viewing window $[-1,000, 1,000]$, the domain is from $-1,000$ to $1,000$. This means we'll evaluate the cube root function for all these $x$-values.

3Step 3: Determine the range

The cube root function, $ y = \sqrt[3]{x} $, is always defined for all real numbers $x$ and outputs all real numbers. Therefore, the range of this function over any domain is all real numbers. However, in practical graphing on a calculator, the visual range would be constrained by the graphing window's height limits.

4Step 4: Graph the function

To graph $ y = \sqrt[3]{x} $ from $x = -1,000$ to $x = 1,000$, plot a smooth curve passing through the origin $(0, 0)$ extending both to the left and to the right. The graph would look like an elongated S-shape, mirroring over the x-axis and y-axis. It increases slowly for negative $x$-values and steadily for positive $x$-values.

Key Concepts

Domain of the Cube Root FunctionRange of the Cube Root FunctionGraphing the Cube Root Function

Domain of the Cube Root Function

The domain of a function refers to all the possible input values (usually denoted as $x$) that can be used in the function to yield a valid output. In the case of the cube root function $y = \sqrt[3]{x}$, the domain is all real numbers. This is because you can take the cube root of any real number, whether it's positive, negative, or zero. For our specific exercise, we are looking at the domain defined by the interval $[-1,000, 1,000]$. This means we'll consider every real number between $-1,000$ and $1,000$, inclusive. As you can imagine, that's quite a broad range! The cube root function will smoothly and continuously process each of these $x$ values without restriction, emphasizing the comprehensiveness of its domain.In summary:

The domain for $y = \sqrt[3]{x}$ is all real numbers: $(-\infty, \infty)$.
For our specific graph window, the domain is $[-1,000, 1,000]$.
No values are excluded from the domain in any range selected for this function.

Range of the Cube Root Function

The range of a function represents all the potential outputs that can result when all the possible values of the domain are input into the function. For the cube root function $y = \sqrt[3]{x}$, the range is similarly all real numbers. This means for any real number $x$ you pick from the function's domain, $y$ can be any real number.What's fascinating about $y = \sqrt[3]{x}$ is that it isn't limited to non-negative values, unlike the square root function which only outputs zero or positive numbers. To illustrate, here are a few outcomes:

$x = 0$ results in $y = 0$
A positive $x$ like 8 will result in a positive $y$ such as $2$
A negative $x$ like $-8$ will correspondingly give a negative $y$ like $-2$

Thus, even within our chosen graphing window, the true range remains all real numbers. While graphing calculators and windows might display limits visually, the actual mathematical range remains infinite and unbounded, encompassing all real numbers.

Graphing the Cube Root Function

Graphing functions like $y = \sqrt[3]{x}$ helps visualize how inputs relate to outputs and appreciate their behavior. The cube root function's graph has a distinctive elongated S-shape and is quite symmetrical.Steps for graphing:

Start at the origin $0, 0$, as $\sqrt[3]{0} = 0$.
For $x$-values that are positive, such as 1, 8, or 27, their cube roots (1, 2, and 3 respectively) are calculated smoothly and plotted above the $x$-axis.
For negative $x$-values, like -1, -8, or -27, the corresponding $y$ values (-1, -2, and -3) show the graph dipping below the $x$-axis.
The curve moves leftwards and rightwards symmetrically from the origin, illustrating that the same magnitude of positive and negative inputs produces outputs of equal magnitude but opposite signs.

The entire curve is smooth and continuous, crossing through the origin. It's important to note that though our visual graph is bounded within a certain window $[-1,000, 1,000]$, the real behavior of the function stretches infinitely in both directions. When plotting points, emphasize the steadily increasing nature in both positive and negative directions, maintaining its distinctive mirrored shape across both the x-axis and y-axis.

Problem 86

Problem 87

Other exercises in this chapter

Problem 86

Let $F(x)=(x+1)^{5}, f(x)=x^{5},$ and $g(x)=x+1$. True or False: $(f \circ g)(x)=F(x)$.

View solution

Problem 86

For the following exercises, let $F(x)=(x+1)^{5}, f(x)=x^{5},$ and $g(x)=x+1$. True or False: $(f \circ g)(x)=F(x)$.

View solution

Problem 87

Find the composition when $f(x)=x^{2}+2$ for all $x \geq 0$ and $g(x)=\sqrt{x-2}$. $$ (f \circ g)(6) ;(g \circ f)(6) $$

View solution

Problem 87

For the following exercises, find the composition when $f(x)=x^{2}+2$ for all $x \geq 0$ and $g(x)=\sqrt{x-2}$. $$(f \circ g)(6) ;(g \circ f)(6)$$

View solution