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

Verified step by step guidance

1. 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.
2. 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.
3. 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}.
4. 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.
5. 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.

Key Concepts

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.