Problem 22 - College Algebra and Calculus: An Applied Approach

Question

Describe the right-hand and left-hand behavior of the graph of the polynomial function. $$h(x)=1-x^{6}$$

Step-by-Step Solution

Answer

The right-hand behavior of the function \( h(x) = 1 - x^{6} \) is that it falls to negative infinity as \( x \) approaches positive infinity. The left-hand behavior of the function is that it also falls to negative infinity as \( x \) approaches negative infinity.

Step 1: Identifying the Polynomial Function

First, identify the polynomial function given in the question. Here, the function is \( h(x) = 1 - x^{6} \).

Step 2: Find the Right-hand behavior

The Right-hand behavior refers to what happens to the function values \( h(x) \) as \( x \) approaches positive infinity. In this function, as \( x \) approaches infinity, \( x^{6} \) becomes infinitely large. Therefore, \( 1 - x^{6} \) will approach negative infinity. So, the graph of the function falls to negative infinity as \( x \) approaches positive infinity.

Step 3: Find the Left-hand behavior

The Left-hand behavior defines what happens to the function values \( h(x) \) as \( x \) approaches negative infinity. In this function, as \( x \) approaches negative infinity, \( x^{6} \) becomes infinitely large because any negative number raised to an even power results in a positive number. So, \( 1 - x^{6} \) approaches negative infinity as well. Therefore, the graph of the function falls to negative infinity as \( x \) approaches negative infinity as well.