Ch. 2 - Graphs and Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 2 - Graphs and Functions

Problem 87a

Chapter 3, Problem 87a

Determine the largest open intervals of the domain over which each function is (a) increasing. See Example 9.

Verified step by step guidance

Identify the critical points on the graph where the function changes from increasing to decreasing or vice versa. In this graph, the critical points are at $x = -3$ and $x = 0$.

Recall that a function is increasing on intervals where the graph moves upward as $x$ increases. Look at the graph to determine where the function rises as you move from left to right.

From the graph, observe that the function increases from the far left (negative infinity) up to the point $(-3, 5)$. This means the function is increasing on the interval $(-\infty, -3)$.

After $x = -3$, the function starts to decrease until $x = 0$. At $x = 0$, the function becomes constant (horizontal line), so it is neither increasing nor decreasing beyond this point.

Therefore, the largest open interval where the function is increasing is $(-\infty, -3)$. Write this interval as your answer.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Increasing and Decreasing Intervals

A function is increasing on an interval if, as x moves from left to right, the function values rise. Conversely, it is decreasing if the function values fall. Identifying these intervals involves analyzing the graph or the derivative to see where the slope is positive or negative.

Open Intervals

Open intervals are intervals that do not include their endpoints. When determining where a function is increasing, we focus on the largest open intervals where the function consistently rises, excluding points where the function changes direction or is constant.

Critical Points and Local Extrema

Critical points occur where the function's derivative is zero or undefined, often corresponding to local maxima or minima. These points mark where the function changes from increasing to decreasing or vice versa, helping to identify the boundaries of increasing intervals.

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Related Practice

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Given functions f and g, find (ƒ∘g)(x) and its domain. ƒ(x)=1/(x-2), g(x)=1/x

Textbook Question

Determine the largest open intervals of the domain over which each function is b) decreasing. See Example 9.

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Textbook Question

Given functions f and g, (g∘ƒ)(x) and its domain. ƒ(x)=1/(x-2), g(x)=1/x

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Textbook Question

Graph each function. See Examples 6–8 and the Summary of Graphing Techniques box following Example 9. ƒ(x)=3√x-2

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