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Chapter 3: Problem 83

Use a graphing utility to graph each pair of functions in the same viewing window. Describe any similarities and differences in the graphs. (a) \(y_{1}=2^{x}, y_{2}=x^{2}\) (b) \(y_{1}=3^{x}, y_{2}=x^{3}\)

Short Answer

Expert verified
For the function pairs \(y_{1}=2^{x}\) and \(y_{2}=x^{2}\), both functions yield non-negative y-values, but the exponential function is always increasing unlike the quadratic function, and also has the x-axis as an asymptote, which \(x^{2}\) doesn't. For the pair \(y_{1}=3^{x}\) and \(y_{2}=x^{3}\), both functions cross through the point (0,1) and increase for positive x, but \(x^{3}\) decreases for negative x while \(3^{x}\) approaches 0. These differences and similarities arise due to the individual characteristics of the functions and their behaviors for various x_values.

Step by step solution

01

Graph the Given Function Pairs

Begin by graphing the first pair of functions \(y_{1}=2^{x}\) and \(y_{2}=x^{2}\) in the same viewing window. Repeat the same process for the second pair \(y_{1}=3^{x}\) and \(y_{2}=x^{3}\). After graphing these functions, observe their shapes and crucial points.
02

Describe Similarities and Differences

For the first pair, both \(y_{1}=2^{x}\) and \(y_{2}=x^{2}\) have the graph of all non-negative y-values. However, the exponential function \(2^{x}\) is always increasing, while the quadratic function \(x^{2}\) has a minimum turning point at x=0. Moreover, \(2^{x}\) has the x-axis as a horizontal asymptote, but \(x^{2}\) does not have an asymptote. For the second pair, \(y_{1}=3^{x}\) is an increasing function and \(y_{2}=x^{3}\) changes sign and grows faster than \(3^{x}\) for large enough values. Both functions cross through the point (0,1) and increase for positive x, but \(x^{3}\) decreases for negative x whereas \(3^{x}\) approaches 0.
03

Conclude the Comparisons

From the differences and similarities, it's noted that the graphs of the functions exhibit unique characteristics related to their form. While there are some similarities like both functions being positive for positive x, the differences are primarily due to the functions' nature, with exponential functions always increasing while polynomial functions can have maximum or minimum turning points.

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