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Chapter 1: Problem 51

(a) use a graphing utility to graph the function and visually determine the intervals over which the function is increasing, decreasing, or constant, and (b) make a table of values to verify whether the function is increasing, decreasing, or constant over the intervals you identified in part (a). $$ f(t)=-t^{4} $$

Short Answer

Expert verified
The function \(f(t) = -t^4\) is decreasing over the interval \(-\infty < t < \infty\).

Step by step solution

01

Graph the Function

Using a graphing utility, the function \(f(t) = -t^4\) should be graphed. When examining the graph, focus on looking for intervals where the function is increasing, decreasing, or constant.
02

Determine Intervals from the Graph

On careful observation, it can be seen that the function decreases as it moves from left to right. Therefore, we can conclude that the function is decreasing over the interval \(-\infty < t < \infty\), and it is neither increasing nor constant anywhere.
03

Make a Table of Values

Create a table with several values of \(t\) and their corresponding \(f(t)\) values. This can confirm whether the function is decreasing over the interval. The table may contain values from both negative and positive integers for \(t\). The \(f(t)\) values should get smaller as you move from left to right in the table, confirming the function is decreasing.

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Determine whether the function has an inverse function. If it does, find the inverse function. $$ f(x)=(x+3)^{2}, \quad x \geq-3 $$
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