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Chapter 1: Problem 6

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

Short Answer

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#short_answer# The graph of \(y = x^{1/5}\) has the following properties: - Domain and Range: Both are all real numbers (\(\mathbb{R}\)) - Symmetry: The graph has symmetry about the \(y\)-axis. - Behavior as \(x\to\pm\infty\): As \(x\) approaches positive or negative infinity, \(y\) also approaches positive or negative infinity, respectively. - Critical Point: There is a critical point at \((0,0)\). - Points of Intersection with the Axes: The graph intersects both the \(x\)-axis and the \(y\)-axis at the origin \((0,0)\). The graph is a smooth curve with a flat slope at the origin that extends indefinitely in the positive and negative directions along both the \(x\) and \(y\) axes.

Step by step solution

01

Identify the Domain and Range

For this function, the domain is the set of all possible input values of \(x\). Since we are dealing with a fractional exponent, we can use any real number as the input. Therefore, the domain will be all real numbers (denoted by \(\mathbb{R}\)). The range of the function consists of all possible output values, or the set of all \(y\) values that the function can produce. Since \(x^{1/5}\) represents the fifth root of \(x\), the possible output values can also be any real number. Hence, the range is also all real numbers (denoted by \(\mathbb{R}\)).
02

Analyze the Symmetry

To analyze the symmetry of the graph, we can replace \(x\) with \(-x\) and see if the resulting function is the same: \[y = (-x)^{1/5} = x^{1/5}.\] Since the function remains the same, the graph has symmetry about the \(y\)-axis.
03

Determine the Behavior as x Approaches Infinity

As \(x\) approaches positive or negative infinity, we want to analyze how the function behaves. When \(x\) approaches positive infinity, i.e., \(x\to\infty\), we have: \[y = x^{1/5} \to\infty,\] so the graph extends to positive infinity in the \(y\) direction. Similarly, as \(x\) approaches negative infinity, i.e., \(x\to-\infty\), we have: \[y = x^{1/5} \to-\infty,\] so the graph extends to negative infinity in the \(y\) direction.
04

Identify Any Critical Points

Critical points occur when the derivative of the function is equal to zero or does not exist. For this function, the derivative is: \[\frac{dy}{dx}=\frac{1}{5}x^{-4/5}.\] The derivative does not exist at \(x=0\), so there is a critical point at \((0,0)\).
05

Identify Points of Intersection with the Axes

The function intersects the \(x\)-axis when \(y=0\). In this case, we have: \[0 = x^{1/5}.\] Since the function has an odd root, there will be one real root for this equation, which is \(x=0\). The function intersects the \(y\)-axis when \(x=0\). In this case, we have: \[y = 0^{1/5} = 0.\] So the function intersects the \(y\)-axis at \((0,0)\), which is the same point as the \(x\)-axis intersection.
06

Sketch the Graph

Based on our analysis, we know that the graph has symmetry about the \(y\)-axis, has a critical point at \((0,0)\), and intersects both axes at \((0,0)\). As \(x\to\pm\infty\), \(y\to\pm\infty\). Using these key features, we can sketch the graph of \(y = x^{1/5}\) as a smooth curve with a flat slope at the origin that extends indefinitely in the positive and negative directions along both the \(x\) and \(y\) axes.

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Key Concepts

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

Domain and Range
When analyzing functions, understanding the domain and range is essential. The domain of a function refers to all the possible values that can be input into the function, essentially covering all values of the independent variable, typically denoted as \(x\). In the function \(y = x^{1/5}\), the expression is defined for any real number \(x\). Hence, the domain includes all real numbers, represented as \(\mathbb{R}\).
Understanding the range, on the other hand, involves determining all potential output values, or \(y\) values, that result from substituting domain values into the function. Because \(x^{1/5}\) is the fifth root of \(x\), it can produce any real number. Thus, similar to the domain, the range is also all real numbers, expressed as \(\mathbb{R}\).
To summarize:
  • Domain: all real numbers (\(\mathbb{R}\))
  • Range: all real numbers (\(\mathbb{R}\))
Symmetry of Functions
Symmetry in functions describes situations where different transformations result in the same graph. To determine if the function \(y = x^{1/5}\) is symmetric, we replace \(x\) with \(-x\). Calculating:
  • Replace \(y = (-x)^{1/5}\)
  • This simplifies to \(y = x^{1/5}\)
Since the transformed function is the same, \(y = x^{1/5}\) exhibits symmetry about the y-axis, making it an even function. This type of symmetry means the graphical representation is identical on either side of the y-axis. It's like having a mirror along the y-axis.
Recognizing symmetry can be beneficial when sketching or analyzing graphs, as it helps predict their behavior across different sections of the x-axis efficiently.
Critical Points
Critical points of a function are important as they can indicate points of interest on the graph, such as maxima, minima, or points where the function changes direction. To find these points, we use the derivative of the function. For \(y = x^{1/5}\), the derivative is:
  • \(\frac{dy}{dx} = \frac{1}{5}x^{-4/5}\)
This derivative does not exist when \(x = 0\), indicating a critical point at \((0,0)\). A critical point might not always lead to a local maximum or minimum, but it marks a special point in the function. In this case, the result at the origin reveals the function's graph has a flat slope here, contributing to its smooth extension.
It's essential to identify critical points, as these provide insights into the function's graph structure and can assist in understanding its overall behavior.
Intersection with Axes
Finding where a graph intersects the axes gives crucial information about the function's behavior. Intersection with the x-axis occurs when \(y = 0\). For our function, this means:
  • \(0 = x^{1/5}\)
The only solution is \(x = 0\), leading to an intersection at \((0, 0)\).
Similarly, intersection with the y-axis occurs when \(x = 0\):
  • Calculating, \(y = 0^{1/5} = 0\)
This also provides an intersection at \((0, 0)\).
Thus, for \(y = x^{1/5}\), both axes meet the curve at the origin. Understanding intersections is valuable for sketching the graph, as these points give reference adjustments, assisting in accurately plotting or predicting the graph's path.

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