Increasing functions - Examples, Exercises and Solutions

Question Types:
Increasing and Decreasing Intervals of a Function: Identifying which domain is applicable for the described functionIncreasing and Decreasing Intervals of a Function: Identifying increase and decrease from a verbal description of the functionIncreasing and Decreasing Intervals of a Function: Matching the graph to the storyIncreasing and Decreasing Intervals of a Function: Determine whether or not it is possible to create the functionIncreasing and Decreasing Intervals of a Function: Finding Increasing or Decreasing Domains

Increasing functions

What is an Increasing Function?

An increasing function is a type of relationship where, as you move to the right on the graph (increasing the xx-value), the yy-value also gets bigger. It’s like climbing a hill—the higher you go (the more you increase xx), the more your height (the yy-value) increases.

We will say that a function is increasing when, as the value of the independent variable X X increases, the value of the function Y Y increases.

How to Spot an Increasing Function:

  1. On a Graph: The line or curve goes upwards as you move from left to right.
  2. In Numbers: For any two xxx-values, if the second number is larger than the first x2>x1​x_2 > x_1​, then the second yy-value will also be larger than the first f(x2)>f(x1)f(x_2) > f(x_1).

Real-Life Example:

Think about saving money in a piggy bank. Every day you add more coins, and the total amount of money keeps going up. That’s an increasing function in action—your savings are the yy-values, and the number of days is the xx-values.

Fun Fact:

If the line or curve never stops going up, it's called strictly increasing. If it flattens for a bit before going up again, it's just increasing.

let's see an example of strictly increasing linear function:

increasing function

Detailed explanation

Suggested Topics to Practice in Advance

  1. Ways to Represent a Function
  2. Representing a Function Verbally and with Tables
  3. Graphical Representation of a Function
  4. Algebraic Representation of a Function
  5. Notation of a Function
  6. Rate of Change of a Function
  7. Variation of a Function
  8. Rate of change represented with steps in the graph of the function
  9. Rate of change of a function represented graphically
  10. Constant Rate of Change
  11. Variable Rate of Change
  12. Rate of Change of a Function Represented by a Table of Values

Practice Increasing functions

Question 1

In what domain does the function increase?

000

Question 2

In what domain does the function increase?

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

Question 3

In what domain is the function increasing?

–5–5–5555101010151515–5–5–5555000

Question 4

In what domain is the function negative?

–0.5–0.5–0.50.50.50.51111.51.51.5222000

Question 5

In what interval is the function increasing?

Purple line: \( x=0.6 \)

111222333111000

Examples with solutions for Increasing functions

Exercise #1

In what domain does the function increase?

000

Video Solution

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 , meaning the Y values are increasing.

Answer

x<0

Exercise #2

In what domain does the function increase?

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

Video Solution

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 meaning the Y values are increasing.

Answer

x > 0

Exercise #3

In what domain is the function increasing?

–5–5–5555101010151515–5–5–5555000

Video Solution

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

Exercise #4

In what domain is the function negative?

–0.5–0.5–0.50.50.50.51111.51.51.5222000

Video Solution

Step-by-Step Solution

Let's 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 see that in the domain x > 1 the function is decreasing, meaning the Y values are decreasing.

Answer

x > 1

Exercise #5

In what interval is the function increasing?

Purple line: x=0.6 x=0.6

111222333111000

Video Solution

Step-by-Step Solution

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

Answer

x<0.6

Question 1

Determine the domain of the following function:

A function describing the charging of a computer battery during use.

Question 2

Determine the domain of the following function:

The function describes a student's grades throughout the year.

Question 3

Determine the domain of the following function:

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

Question 4

Determine which domain corresponds to the described function:

The function describes a person's energy level throughout the day.

Question 5

Determine which domain corresponds to the function described below:

The function represents the amount of fuel in a car's tank according to the distance traveled by the car.

Exercise #6

Determine the domain of the following function:

A function describing the charging of a computer battery during use.

Step-by-Step Solution

According to logic, the computer's battery during use will always decrease since the battery serves as an energy source for the computer.

Therefore, the domain that suits this function is - always decreasing.

Answer

Always decreasing

Exercise #7

Determine the domain of the following function:

The function describes a student's grades throughout the year.

Step-by-Step Solution

According to logic, the student's grades throughout the year depend on many criteria that are not given to us.

Therefore, the appropriate domain for the function is - it is impossible to know.

Answer

Impossible to know.

Exercise #8

Determine the domain of the following function:

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

Step-by-Step Solution

Logically, a person's weight is something that fluctuates.

In one week, a person's weight can increase, but in the following week, it can decrease.

Therefore, the domain that suits this function is - partly increasing and partly decreasing.

Answer

Partly increasing and partly decreasing.

Exercise #9

Determine which domain corresponds to the described function:

The function describes a person's energy level throughout the day.

Step-by-Step Solution

Logically, a person's energy level throughout the day changes and depends on many factors.

On one hand, energy levels rise for example when a person eats or consumes caffeine.

On the other hand, energy levels drop when for example a person moves a lot or exercises.

Therefore, the domain that fits this function is - partly increasing and partly decreasing.

Answer

Partly increasing and partly decreasing

Exercise #10

Determine which domain corresponds to the function described below:

The function represents the amount of fuel in a car's tank according to the distance traveled by the car.

Step-by-Step Solution

According to the definition, the amount of fuel in the car's tank will always decrease, since during the trip the car consumes fuel in order to travel.

Therefore, the domain that is suitable for this function is - always decreasing.

Answer

Always decreasing

Question 1

Determine which domain corresponds to the function described below:

The function represents the height of a child from birth to first grade.

Question 2

In which interval does the function decrease?

Red line: \( x=0.65 \)

111222333–1–1–1111000

Question 3

In what domain does the function increase?

Black line: \( x=1.1 \)

–2–2–2222444666222000

Question 4

In what domain does the function increase?

Green line:
\( x=-0.8 \)

–2–2–2222222000

Question 5

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.

Exercise #11

Determine which domain corresponds to the function described below:

The function represents the height of a child from birth to first grade.

Step-by-Step Solution

According to logic, a child's height from birth until first grade will always be increasing as the child grows.

Therefore, the domain that suits this function is - always increasing.

Answer

Always increasing.

Exercise #12

In which interval does the function decrease?

Red line: x=0.65 x=0.65

111222333–1–1–1111000

Video Solution

Step-by-Step Solution

Remember that a function is increasing if both the x x values and the y y values are increasing simultaneously.

A function is decreasing if the x x values are increasing while the y y values are decreasing simultaneously.

In the graph we can see that the function is decreasing in all domains. In other words, it is decreasing for all x x .

Answer

All values of x x

Exercise #13

In what domain does the function increase?

Black line: x=1.1 x=1.1

–2–2–2222444666222000

Video Solution

Step-by-Step Solution

Remember that a function is increasing if the x x values and y y values are increasing simultaneously.

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

In the plotted graph, we can see that in the domain 1.1 > x > 0 the function is increasing—meaning the y y values are increasing.

Answer

1.1 > x > 0

Exercise #14

In what domain does the function increase?

Green line:
x=−0.8 x=-0.8

–2–2–2222222000

Video Solution

Step-by-Step Solution

The function increases if X values and Y values increase simultaneously.
In this function, despite its unusual form, we can see that the function continues to increase according to the definition at all times,
and there is no stage where the function decreases.
Therefore, we can say that the function increases for all X, there is no X we can input where the function will be decreasing.

Answer

All values of x x

Exercise #15

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.

Step-by-Step Solution

According to logic, the speed of the stone during a fall from a great height will increase as it falls with acceleration.

In other words, the speed of the stone increases, so the appropriate domain for this function is - always increasing.

Answer

Always increasing

More Questions

Topics learned in later sections

  1. Functions for Seventh Grade
  2. Increasing and Decreasing Intervals (Functions)
  3. Decreasing function
  4. Constant Function
  5. Decreasing Interval of a function
  6. Increasing Intervals of a function
  7. Domain of a Function
  8. Indefinite integral
  9. Inputing Values into a Function