Decreasing Intervals Practice: Function Analysis Problems

Master identifying decreasing intervals of functions with step-by-step practice problems. Learn where function values decrease as x increases through guided exercises.

📚What You'll Practice with Decreasing Intervals
  • Identify decreasing intervals where function values drop as x increases
  • Locate maximum points that begin decreasing intervals in graphs
  • Analyze function behavior using interval notation for decreasing sections
  • Distinguish between increasing and decreasing intervals on function graphs
  • Apply decreasing interval concepts to real-world function problems
  • Interpret function values and their relationship to x-value changes

Understanding Decreasing Interval of a function

Complete explanation with examples

The decreasing intervals of a function

A decreasing interval of a function expresses the same values of X (the interval), in which the values of the function (Y) decrease parallelly to the increase of the values of X to the right.

In certain cases, the decreasing interval begins at the maximum point, but it does not necessarily have to be this way.

Detailed explanation

Practice Decreasing Interval of a function

Test your knowledge with 17 quizzes

Determine the domain of the following function:

The function represents the weight of a person over a period of 3 years.

Examples with solutions for Decreasing Interval of a function

Step-by-step solutions included
Exercise #1

Determine in which domain the function is negative?

–0.5–0.5–0.50.50.50.51111.51.51.5222000

Step-by-Step Solution

Remember that a function is increasing if both X values and Y values are increasing simultaneously.

A function is decreasing if X values are increasing while Y values are decreasing simultaneously.

In the graph, we can observe that in the domain x>1 x > 1 the function is decreasing, meaning the Y values are decreasing.

Answer:

x>1 x > 1

Video Solution
Exercise #2

Does the function in the graph decrease throughout?

YYYXXX

Step-by-Step Solution

To solve this problem, we'll begin by examining the graph of the function provided:

  • Step 1: Observe the graph from left to right along the x-axis.
  • Step 2: Look for any intervals where the function value (y-coordinate) does not decrease as the x-value increases.
  • Step 3: Pay special attention to segments where the graph might look horizontal or rising.

Upon inspecting the graph, we find:

- There are sections where the function's y-values appear to remain constant or potentially rise as the x-values increase. Specifically, even if the function decreases in major portions, any interval where it doesn't means the function cannot be classified as decreasing throughout.

Thus, the function does not strictly decrease on the entire interval shown. Therefore, the solution to the problem is No.

Answer:

No

Exercise #3

In what domain does the function increase?

–20–20–20–10–10–10101010202020–10–10–10101010000

Step-by-Step Solution

Let's remember that the function increases if the x x values and y y values increase simultaneously.

On the other hand, the function decreases if the x x values increase while the y y values decrease simultaneously.

In the given graph, we can see that the function increases in the domain where x>0 x > 0 ; in other words, where the y y values are increasing.

Answer:

x>0 x > 0

Video Solution
Exercise #4

In what domain does the function increase?

000

Step-by-Step Solution

Let's remember that the function increases if the X values and Y values increase simultaneously.

On the other hand, the function decreases if the X values increase and the Y values decrease simultaneously.

In the given graph, we notice that the function increases in the domain where x<0 x < 0 , meaning the Y values are increasing.

Answer:

x<0 x<0

Video Solution
Exercise #5

In what domain is the function increasing?

–5–5–5555101010151515–5–5–5555000

Step-by-Step Solution

Let's first remember that a function is increasing if both the X and Y values are increasing simultaneously.

Conversely, a function is decreasing if the X values are increasing while the Y values are decreasing simultaneously.

In the graph shown, we can see that the function is increasing in every domain and therefore the function is increasing for all values of X.

Answer:

All values of x x

Video Solution

Frequently Asked Questions

Everything you need to know about Decreasing Interval of a function

What is a decreasing interval of a function?

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A decreasing interval is a range of x-values where the function values (y-values) get smaller as x increases from left to right. On a graph, this appears as a downward slope moving from left to right.

How do you find decreasing intervals on a graph?

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Look for sections where the graph slopes downward from left to right. These are the decreasing intervals. You can identify them by checking: 1) Find where the graph goes down, 2) Note the x-values at the start and end, 3) Write the interval using proper notation.

Do decreasing intervals always start at maximum points?

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No, decreasing intervals don't always begin at maximum points. While they often start after a peak (maximum), they can also begin at the start of the function's domain or after other types of critical points.

What's the difference between increasing and decreasing intervals?

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Increasing intervals show function values rising as x increases (upward slope), while decreasing intervals show function values falling as x increases (downward slope). They represent opposite behaviors of the function.

How do you write decreasing intervals in interval notation?

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Use parentheses or brackets to show the x-values where the function decreases. For example, if a function decreases from x = 2 to x = 5, write it as (2, 5) or [2, 5] depending on whether the endpoints are included.

Can a function have multiple decreasing intervals?

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Yes, functions can have multiple decreasing intervals separated by increasing sections or constant regions. Each separate downward-sloping section represents a distinct decreasing interval that should be identified separately.

Why are decreasing intervals important in math?

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Decreasing intervals help analyze function behavior, predict trends, and solve optimization problems. They're essential for understanding rates of change, finding minimum values, and modeling real-world situations like declining populations or decreasing temperatures.

What are common mistakes when identifying decreasing intervals?

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Common errors include: confusing x-intervals with y-values, mixing up increasing and decreasing directions, forgetting to use proper interval notation, and not recognizing that steep downward slopes are still decreasing intervals regardless of steepness.

More Decreasing Interval of a function Questions

Click on any question to see the complete solution with step-by-step explanations

Increasing and Decreasing Intervals of a Function

Finding Decreasing Intervals: Analyzing Function Behavior from Graph Identifying Decreasing Domains in a Piecewise Function Graph Finding Decreasing Intervals on a Curved Function Graph: [0,6] Analysis Finding Function Increase Domain with Vertical Line x=-0.8: Graph Analysis Finding Decreasing Intervals: Graph Analysis with x=0.65 Reference Line

Continue Your Math Journey

Master Decreasing Interval of a function first, then advance to these related topics that build on your fraction skills

Suggested Topics to Practice in Advance

Ways to Represent a FunctionRepresenting a Function Verbally and with TablesGraphical Representation of a FunctionAlgebraic Representation of a FunctionNotation of a FunctionRate of Change of a FunctionVariation of a FunctionRate of change represented with steps in the graph of the functionRate of change of a function represented graphicallyConstant Rate of ChangeVariable Rate of ChangeRate of Change of a Function Represented by a Table of Values

Topics Learned in Later Sections

Functions for Seventh GradeIncreasing and Decreasing Intervals (Functions)Increasing functionsDecreasing functionConstant FunctionIncreasing Intervals of a functionDomain of a FunctionIndefinite integralInputing Values into a Function

Practice by Question Type

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