Examples with solutions for Increasing and Decreasing Intervals of a Function: Identifying increase and decrease from a verbal description of the function

Exercise #1

Determine whether the function is increasing, decreasing, or constant. For each function check your answers with a graph or table.

For each number, multiply by(1) (-1) .

Video Solution

Step-by-Step Solution

The function is:

f(x)=(1)x f(x)=(-1)x

Let's start by assuming that x equals 0:

f(0)=(1)×0=0 f(0)=(-1)\times0=0

Now let's assume that x equals minus 1:

f(1)=(1)×(1)=1 f(-1)=(-1)\times(-1)=1

Now let's assume that x equals 1:

f(1)=(1)×1=1 f(1)=(-1)\times1=-1

Now let's assume that x equals 2:

f(2)=(1)×2=2 f(2)=(-1)\times2=-2

Let's plot all the points on the function graph:

–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666–3–3–3–2–2–2–1–1–1111222000

We can see that the function we got is a decreasing function.

Answer

Decreasing

Exercise #2

Determine whether the function is increasing, decreasing, or constant. Check your answer with a graph or table.

For each number corresponding to the number 1 -1 .

Video Solution

Step-by-Step Solution

The function is:

f(x)=x1 f(x)=x-1

Let's start with x equal to 0:

f(0)=01=1 f(0)=0-1=-1

Now let's assume x is equal to 1:

f(1)=11=0 f(1)=1-1=0

Now let's assume x is equal to 2:

f(2)=21=1 f(2)=2-1=1

Now let's assume x is equal to minus 1:

f(1)=(1)1=2 f(-1)=(-1)-1=-2

Let's plot all the points on the function graph:

–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666–3–3–3–2–2–2–1–1–1111222000

It appears that the function we obtained is an increasing function.

Answer

Increasing

Exercise #3

Determine whether the function is increasing, decreasing, or constant. For each function check your answers with a graph or table.

For each number, multiply by 0.

Video Solution

Step-by-Step Solution

The function is:

f(x)=x×0 f(x)=x\times0

Let's start by assuming that x equals 0:

f(0)=0×0=0 f(0)=0\times0=0

Now let's assume that x equals 1:

f(1)=1×0=0 f(1)=1\times0=0

Now let's assume that x equals -1:

f(1)=(1)×0=0 f(-1)=(-1)\times0=0

Now let's assume that x equals 2:

f(2)=2×0=0 f(2)=2\times0=0

Let's plot all the points on the function's graph:

–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666–3–3–3–2–2–2–1–1–1111222000

We can see that the function we obtained is a constant function.

Answer

Constant

Exercise #4

Determine whether the function is increasing, decreasing, or constant. For each function check your answers with a graph or table:

Each number is divided by (1) (-1) .

Video Solution

Step-by-Step Solution

The function is:

f(x)=x1 f(x)=\frac{x}{-1}

Let's start by assuming that x equals 0:

f(0)=01=0 f(0)=\frac{0}{-1}=0

Now let's assume that x equals 1:

f(1)=11=1 f(1)=\frac{1}{-1}=-1

Now let's assume that x equals 2:

f(1)=11=1 f(-1)=\frac{-1}{-1}=1

Let's plot all the points on the function graph:

–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666–1–1–1111222333444000

We see that we got a decreasing function.

Answer

Decreasing

Exercise #5

Determine whether the function is increasing, decreasing, or constant. For each function check your answers with a graph or table.

Each number is multiplied by the same number with different signs

Video Solution

Step-by-Step Solution

The function is:

f(x)=x×(x) f(x)=x\times(-x)

Let's start by assuming x equals 0:

f(0)=0×0=0 f(0)=0\times0=0

Now let's assume x equals 1:

f(1)=1×1=1 f(1)=1\times-1=-1

Now let's assume x equals minus 1:

f(1)=(1)×(1)=1 f(-1)=(-1)\times(-1)=1

Now let's assume x equals 2:

f(2)=2×(2)=4 f(2)=2\times(-2)=-4

Now let's assume x equals minus 2:

f(2)=(2)×(2)=4 f(-2)=(-2)\times(-2)=4

Let's plot all the points on the function graph:

–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111000

We can see that the function we obtained rises and falls.

Answer

Increasing and decreasing

Exercise #6

Determine whether the function described below is increasing, decreasing, or constant.

Each number is multiplied by the same number.

Video Solution

Step-by-Step Solution

The function is:

f(x)=x2 f(x)=x^2

Let's start with x equal to 0:

f(0)=02=0 f(0)=0^2=0

Now let's assume x is equal to 1:

f(1)=12=1 f(1)=1^2=1

Now let's assume x is equal to 2:

f(2)=22=4 f(2)=2^2=4

Now let's assume x is equal to minus 1:

f(1)=(1)2=1 f(-1)=(-1)^2=1

Now let's assume x is equal to minus 2:

f(2)=(2)2=4 f(-2)=(-2)^2=4

Let's plot all the points on the function graph:

–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666–1–1–1111222333444000

We can see that we got a function that is both increasing and decreasing.

Answer

Increasing and decreasing

Exercise #7

Determine whether the function is increasing, decreasing, or constant. For each function, check your answers using a graph or a table:
Each number is multiplied by 0.5.

Video Solution

Step-by-Step Solution

The function is:

f(x)=0.5x f(x)=0.5x

Let's start with x equal to 0:

f(0)=0×0.5=0 f(0)=0\times0.5=0

Now let's assume x is equal to 1:

f(1)=1×0.5=0.5 f(1)=1\times0.5=0.5

Now let's assume x is equal to 2:

f(2)=2×0.5=1 f(2)=2\times0.5=1

Let's record all the data in a table:

X01200.51f(x)Note that the function is always increasing.

Answer

Growing

Exercise #8

Determine whether the function is increasing, decreasing, or constant. For each function check your answers using a graph or a table. Each number is multiplied by itself three times.

Video Solution

Step-by-Step Solution

The function is:

f(x)=xxx f(x)=xxx

Let's start with x equal to 0:

f(0)=0×0×0=0 f(0)=0\times0\times0=0

Now let's assume x is equal to 1:

f(1)=1×1×1=1 f(1)=1\times1\times1=1

Now let's assume x is equal to 2:

f(2)=2×2×2=8 f(2)=2\times2\times2=8

Let's record all the data in a table:

X012018f(x)Note that the function is always increasing.

Answer

Growing