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.