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Chapter 1: Problem 6

Sketch a graph of \(y=x^{1 / 5}\).

Short Answer

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Question: Based on the given information, sketch the graph of the function \(y=x^{1/5}\), and determine its domain and range. Answer: The graph of the function \(y=x^{1/5}\) will resemble a sideways S curve, passing through the key points \((-8,-2)\), \((-1,-1)\), \((0,0)\), \((1,1)\), and \((32,2)\). The domain and range of the function are both \((-\infty, \infty)\).

Step by step solution

01

Determine the domain and range

The domain of a function is the set of all possible x-values for which the function is defined. Since the exponent is \(1/5\), the function is defined for all real numbers as there are no restrictions. This means we can take the fifth root of any positive or negative number. So, the domain of the function is \((-\infty, \infty)\). The range of a function is the set of all possible y-values that the function can take. As x takes all real numbers, the fifth root of x will also span all real numbers. So, the range of the function is \((-\infty, \infty)\).
02

Plot key points on the coordinate plane

In order to properly sketch the graph, it is useful to identify and plot some key points. These points help us understand the general shape and behavior of the graph. We will calculate the y-values for some chosen x-values and plot these points on the coordinate plane. For the x-values, we will pick \(-8, -1, 0, 1\), and \(32\): For \(x=-8\), \(y = (-8)^{1/5} = -2\) For \(x=-1\), \(y = (-1)^{1/5} = -1\) For \(x=0\), \(y = (0)^{1/5} = 0\) For \(x=1\), \(y = (1)^{1/5} = 1\) For \(x=32\), \(y = (32)^{1/5} = 2\) Now we can plot these points on the coordinate plane: \((-8,-2)\), \((-1,-1)\), \((0,0)\), \((1,1)\), and \((32,2)\).
03

Sketch the graph by connecting the points and considering the domain and range

With the key points plotted, we can now sketch the graph of \(y=x^{1/5}\). The graph passes through the origin \((0,0)\) and covers all x-values from negative infinity to positive infinity. The graph is symmetric with respect to the origin, since it is an odd root function. It also grows slowly as we increase the x-value, not as rapidly as a linear or quadratic function would. Connect the plotted points to create the overall shape of the graph, making sure that the graph passes through all the key points and covers the entire domain and range. The final graph should resemble a sideways S curve.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Domain and Range
When we talk about the domain of a function, we are referring to the set of all possible input values (x-values) that the function can accept. For the function given, \(y=x^{1/5}\), any real number is acceptable for the domain. This is because taking the fifth root of both negative and positive numbers is defined, without any restrictions. That makes the domain
  • \((-\infty, \infty)\)

The range is about what the function can output 鈥� the y-values. Since the fifth root can return any real number from an input of any real number, like the input, the output also spans all real numbers. Hence, the range of this function is similarly given by
  • \((-\infty, \infty)\)
Key Points
Key points help in understanding the configuration and progression of a graph. By identifying and plotting these points, you can draw a more accurate graph. For \(y=x^{1/5}\), some valuable x-values are:
  • \(-8\), for which \(y=(-8)^{1/5}=-2\)
  • \(-1\), for which \(y=(-1)^{1/5}=-1\)
  • \(0\), for which \(y=(0)^{1/5}=0\)
  • \(1\), for which \(y=(1)^{1/5}=1\)
  • \(32\), for which \(y=(32)^{1/5}=2\)
These points
  • \((-8,-2)\), \((-1,-1)\), \((0,0)\), \((1,1)\), and \((32,2)\)
are crucial for depicting the graph. They serve as benchmarks, indicating the key transitions where the curve turns or changes its slope.
Coordinate Plane
The coordinate plane is essential for sketching graphs like \(y=x^{1/5}\). This plane is a two-dimensional space where you can plot points expressed as pairs \((x, y)\). Understanding how to use it is critical for graphing any function.
To graph the function:
  • Mark the x-axis (horizontal line) and y-axis (vertical line), which intersect at the origin \((0,0)\).
  • Carefully plot the key points mentioned before on this plane.
  • Connect these points with a smooth curve. This curve represents the function visually.
Since \(y = x^{1/5}\) involves an odd root, the graph will be symmetric with respect to the origin. It forms a sideways S-shaped curve. This symmetry implies that for every positive x-value, there is a negative x-value with a corresponding point mirrored across the origin. Observing the curve on the coordinate plane, note that it extends infinitely along both the x and y axes, complying with its infinite domain and range.

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