Analyzing Cubic Functions: Determining Increasing, Decreasing, or Constant Behavior

Q: Why does the function jump from f(1)=1 to f(2)=8 so quickly?

Cubic functions grow very rapidly because you're multiplying three copies of x together. When x doubles from 1 to 2, the result becomes $ 2 \times 2 \times 2 = 8 $ times larger!

Q: What happens when x is negative in cubic functions?

Great question! Try $ f(-1) = (-1) \times (-1) \times (-1) = -1 $. Since you multiply an odd number of negative values, the result stays negative.

Q: Is this function really always increasing?

Yes, but be careful! The function $ f(x) = x^3 $ is always increasing, meaning as x gets larger, f(x) also gets larger. This includes negative values: -2

Q: How can I verify this with a graph?

Plot points like (-2,-8), (-1,-1), (0,0), (1,1), (2,8) and connect them. You'll see a smooth curve that always goes up from left to right, confirming the function is increasing.

Q: What makes this different from x² functions?

Unlike $ x^2 $ which has a U-shape and decreases then increases, $ x^3 $ functions have an S-shape and are always increasing without any turning points.

Cubic Functions with Basic Value Analysis

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.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Is the function increasing, decreasing or constant?

00:04 Let's check using a table

00:10 Let's substitute X values and calculate the Y values

00:31 We can see that the function is increasing

00:36 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

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.

Step-by-step solution

The function is:

$f(x)=xxx$

Let's start with x equal to 0:

$f(0)=0\times0\times0=0$

Now let's assume x is equal to 1:

$f(1)=1\times1\times1=1$

Now let's assume x is equal to 2:

$f(2)=2\times2\times2=8$

Let's record all the data in a table:

Note that the function is always increasing.

Final Answer

Growing

Key Points to Remember

Essential concepts to master this topic

Cubic Rule: Functions of form $f(x) = x^3$ have different behavior patterns
Test Method: Calculate f(0)=0, f(1)=1, f(2)=8 to see growth pattern
Verification: Create table showing x and f(x) values increasing together ✓

Common Mistakes

Avoid these frequent errors

Testing only positive values
Don't test only x = 0, 1, 2 and conclude function is always increasing! This ignores negative values where cubic functions behave differently. Always test negative values like x = -1, -2 to see the complete behavior pattern.

Practice Quiz

Test your knowledge with interactive questions

Is the function in the graph decreasing?

FAQ

Everything you need to know about this question

Why does the function jump from f(1)=1 to f(2)=8 so quickly?

Cubic functions grow very rapidly because you're multiplying three copies of x together. When x doubles from 1 to 2, the result becomes $2 \times 2 \times 2 = 8$ times larger!

What happens when x is negative in cubic functions?

Great question! Try $f(-1) = (-1) \times (-1) \times (-1) = -1$ . Since you multiply an odd number of negative values, the result stays negative.

Is this function really always increasing?

Yes, but be careful! The function $f(x) = x^3$ is always increasing, meaning as x gets larger, f(x) also gets larger. This includes negative values: -2 < -1 < 0 < 1 < 2.

How can I verify this with a graph?

Plot points like (-2,-8), (-1,-1), (0,0), (1,1), (2,8) and connect them. You'll see a smooth curve that always goes up from left to right, confirming the function is increasing.

What makes this different from x² functions?

Unlike $x^2$ which has a U-shape and decreases then increases, $x^3$ functions have an S-shape and are always increasing without any turning points.

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