Roots and powers Sketch a graph of the given pairs of functions. Be sure to draw the graphs accurately relative to each other.

y = (x)¹⸍³ and y = (x)¹⸍⁵

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Step 1: Understand the functions y = x^{1/3} and y = x^{1/5}. These are both root functions, where y = x^{1/3} is the cube root of x and y = x^{1/5} is the fifth root of x.

Step 2: Analyze the domain and range of both functions. Both functions are defined for all real numbers x, meaning their domain is (-∞, ∞). The range for both functions is also (-∞, ∞) because any real number can be a cube root or fifth root.

Step 3: Consider the behavior of the functions as x approaches positive and negative infinity. As x approaches positive infinity, both y = x^{1/3} and y = x^{1/5} will increase, but y = x^{1/3} will increase faster than y = x^{1/5}. As x approaches negative infinity, both functions will decrease, but again, y = x^{1/3} will decrease faster than y = x^{1/5}.

Step 4: Identify key points to plot. For both functions, when x = 0, y = 0. For x = 1, y = 1 for both functions. For x = -1, y = -1 for both functions. These points will help in sketching the graphs accurately.

Step 5: Sketch the graphs. Start by plotting the key points identified in Step 4. Then, draw smooth curves through these points, ensuring that the graph of y = x^{1/3} is steeper than y = x^{1/5} for both positive and negative values of x.

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주요 개념

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Roots and Powers

Roots and powers are fundamental concepts in algebra and calculus that describe the relationship between numbers and their exponents. A power, such as x raised to a fraction, indicates how many times to multiply x by itself, while a root, like the cube root or fifth root, represents the value that, when raised to a specific power, yields the original number. Understanding these concepts is crucial for analyzing the behavior of functions defined by such expressions.

Graphing Functions

Graphing functions involves plotting points on a coordinate system to visually represent the relationship between the input (x-values) and output (y-values) of a function. For the functions y = x^(1/3) and y = x^(1/5), it is important to understand their shapes, intercepts, and asymptotic behavior. Accurate graphing allows for a comparative analysis of how these functions behave relative to each other, particularly in terms of growth rates and curvature.

Behavior of Radical Functions

Radical functions, such as y = x^(1/3) and y = x^(1/5), exhibit unique characteristics based on their roots. The cube root function is defined for all real numbers and has a point of inflection at the origin, while the fifth root function also spans all real numbers but grows more slowly than the cube root as x increases. Understanding these behaviors is essential for accurately sketching their graphs and comparing their growth and shape.

Roots and powers Sketch a graph of the given pairs of functions. Be sure to draw the graphs accurately relative to each other.y = (x)¹⸍³ and y = (x)¹⸍⁵