Question

Describe the relationship between the graphs of y = x^2 and y = 3√x. How are they alike? How are they different?

Answer 1

Differences:

• x² can be defined over all real numbers, while 3√x only admits positive values for x (or 0). Hence the graph of 3√x starts in x = 0
• 3√x is increasing in its domain, while x² has an interval where it decreases.
• x² has a derivate in every element of its domain, while 3√x doesnt have a derivate in 0
• The second derivate of x² is positive for all its values, while the second derivate of 3√x is negative. Hence, the graph is x² is convex while the graph of 3√x is concave.

Similarities:

• Both functions are increasing for positive values of x
• Both functions have a minimum on the point (0,0)
• The rest of the points of the graph are above the x-axis (this means that both functions take positive values besides 0)
• If you take limit for x reaching +infty, the result is +infty
• The functions doesnt have a horizontal asymptote nor a vertical one