Determine the domain of the function described below: The function represents the amount of water in a pool while it is being filled.

Partly increasing and partly decreasing

Determine the domain of the following function: A function describing the charging of a computer battery during use.

Partly increasing and partly decreasing

Determine the domain of the following function: The function represents the weight of a person over a period of 3 years.

Partly increasing and partly decreasing.

Increasing and Decreasing Function Intervals Practice

Master identifying increasing and decreasing intervals of functions with step-by-step practice problems, graphical analysis, and real-world applications.

📚What You'll Master in This Practice Session

Identify increasing intervals where function values grow as x increases
Locate decreasing intervals where function values fall with increasing x
Analyze function graphs to determine interval notation for increasing/decreasing regions
Connect minimum and maximum points to interval boundaries
Apply interval concepts to real-world situations and word problems
Use proper mathematical notation to express increasing and decreasing intervals

Understanding Increasing Intervals of a function

Complete explanation with examples

The increasing intervals of a function

An increasing interval of a function expresses the same values of X (the interval), in which the values of the function (Y) grow parallel to the growth of the values of X to the right.

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

Detailed explanation

Practice Increasing Intervals of a function

Test your knowledge with 17 quizzes

Which domain corresponds to the described function:

The function represents the velocity of a stone after being dropped from a great height as a function of time.

Examples with solutions for Increasing Intervals of a function

Step-by-step solutions included

Exercise #1

Is the function shown in the graph below decreasing?

Step-by-Step Solution

To solve this problem, we'll follow these steps:

Step 1: Visually inspect the graph to see if it is consistently sloping downward.
Step 2: Apply the definition of a decreasing function.

Now, let's work through each step:
Step 1: Observing the graph, the function's graph is a line moving from the top left to the bottom right. This indicates it slopes downward as we move from left to right across the $x$ -axis.
Step 2: According to the definition of a decreasing function, for any $x_1 < x_2$ , it must hold true that $f(x_1) > f(x_2)$ . Since the graph shows a line moving downward, this condition is satisfied throughout its domain.

Therefore, the function represented by the graph is indeed decreasing.

The final answer is Yes.

Answer:

Yes

Exercise #2

Is the function in the graph decreasing?

Step-by-Step Solution

To analyze whether the function in the graph is decreasing, we must understand how the function's behavior is defined by its graph:

Step 1: Examine the graph. The graph presented is a horizontal line.
Step 2: Recognize the properties of a horizontal line. Horizontally aligned lines correspond to constant functions because the $y$ -value remains the same for all $x$ -values.
Step 3: Define the criteria for a function to be decreasing. A function decreases when, as $x$ increases, the value of $f(x)$ decreases.
Step 4: Apply this criterion to the horizontal line. Since the $y$ -value is constant and does not decrease as $x$ moves rightward, the function is not decreasing.

Therefore, the function represented by the graph is not decreasing.

Answer:

Exercise #3

In what domain is the function increasing?

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 $x$

Video Solution

Exercise #4

In what interval is the function increasing?

Purple line: $x=0.6$

Step-by-Step Solution

Let's remember that a function is described as 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 see that in the domain $x < 0.6$ the function is increasing, meaning the Y values are increasing.

Answer:

$x<0.6$

Video Solution

Exercise #5

Determine in which domain the function is negative?

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 $x > 1$ the function is decreasing, meaning the Y values are decreasing.

Answer:

$x > 1$

Video Solution

Frequently Asked Questions

Everything you need to know about Increasing Intervals of a function

What is an increasing interval of a function?

An increasing interval is a range of x-values where the function values (y) grow as x increases to the right. On a graph, this appears as the function rising from left to right in that interval.

How do you find increasing and decreasing intervals on a graph?

To find intervals: 1) Look at the graph from left to right, 2) Mark where the function rises (increasing) or falls (decreasing), 3) Write intervals using proper notation like (a,b) or [a,b].

Do increasing intervals always start at minimum points?

No, increasing intervals don't necessarily begin at minimum points. While they often do, an increasing interval can start at any point where the function begins to rise.

What's the difference between increasing and decreasing function intervals?

Increasing intervals show function values rising as x increases (upward slope). Decreasing intervals show function values falling as x increases (downward slope).

How do you write interval notation for increasing functions?

Use parentheses ( ) for open intervals and brackets [ ] for closed intervals. For example: increasing on (2,5) means from x=2 to x=5, not including endpoints.

Can a function be both increasing and decreasing?

Yes, most functions have different intervals where they increase and decrease. The function changes behavior at turning points like maximums and minimums.

What are real-world examples of increasing and decreasing intervals?

Examples include: population growth over time (increasing), temperature throughout a day (both increasing and decreasing), or profit vs. production cost relationships in business.

How do maximum and minimum points relate to function intervals?

Maximum points often mark the end of increasing intervals and start of decreasing intervals. Minimum points typically end decreasing intervals and begin increasing intervals.

Continue Your Math Journey

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

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Increasing and Decreasing Function Intervals Practice

Understanding Increasing Intervals of a function

Practice Increasing Intervals of a function

Examples with solutions for Increasing Intervals of a function

Frequently Asked Questions

What is an increasing interval of a function?

How do you find increasing and decreasing intervals on a graph?

Do increasing intervals always start at minimum points?

What's the difference between increasing and decreasing function intervals?

How do you write interval notation for increasing functions?

Can a function be both increasing and decreasing?

What are real-world examples of increasing and decreasing intervals?

How do maximum and minimum points relate to function intervals?

More Increasing Intervals of a function Questions

Increasing and Decreasing Intervals of a Function

Continue Your Math Journey

Suggested Topics to Practice in Advance

Topics Learned in Later Sections

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