[Technology Exercise] Graph the functions in Exercises 113 and 114. Then answer the following questions.


c. How does the graph behave near x = 1 and x = −1?


Give reasons for your answers.


y = (3/2)(x − (1 / x))²/³

Verified step by step guidance

1. Step 1: Begin by understanding the function y = (3/2)(x - (1/x))^(2/3). This is a composite function involving a rational expression inside a fractional power. The expression inside the power is (x - (1/x)), which can be simplified further to understand its behavior.
2. Step 2: Analyze the behavior of the function as x approaches 1 and -1. For x = 1, substitute x = 1 into the expression (x - (1/x)) to get (1 - 1) = 0. Therefore, the function y = (3/2)(0)^(2/3) = 0. This indicates that the graph passes through the point (1, 0).
3. Step 3: For x = -1, substitute x = -1 into the expression (x - (1/x)) to get (-1 - (-1)) = 0. Similarly, the function y = (3/2)(0)^(2/3) = 0. This indicates that the graph also passes through the point (-1, 0).
4. Step 4: Consider the derivative of the function to understand the behavior near x = 1 and x = -1. The derivative will help determine if there are any local maxima, minima, or points of inflection near these values. Calculate the derivative using the chain rule and power rule.
5. Step 5: Examine the limits of the function as x approaches 1 and -1 from both sides. This will help determine if there are any asymptotic behaviors or discontinuities. Consider the limit of (x - (1/x)) as x approaches 1 and -1, and how the fractional power affects the overall function.

Key Concepts

Graphing Functions

Graphing functions involves plotting points on a coordinate plane to visualize the behavior of a function. This helps in understanding how the function behaves at different values of x, including identifying key features such as intercepts, asymptotes, and critical points. Technology tools like graphing calculators or software can assist in accurately plotting complex functions.

Behavior Near Specific Points

Analyzing the behavior of a graph near specific points, such as x = 1 and x = -1, involves examining the function's limits and continuity. This includes determining if the function approaches a particular value, becomes undefined, or exhibits asymptotic behavior. Understanding these behaviors is crucial for predicting the function's behavior in the vicinity of these points.

Fractional Exponents

Fractional exponents, such as ²/³, indicate roots and powers simultaneously. In the function y = (3/2)(x − (1 / x))²/³, the exponent affects the shape and smoothness of the graph. It implies taking the cube root and then squaring the result, which can lead to unique graph characteristics, especially near points where the base expression approaches zero or infinity.