Function Behavior Analysis: Exploring Multiplication by Zero

Q: Why doesn't the function increase when x gets bigger?

Because anything times zero equals zero! Even if x = 1000, we get 1000 × 0 = 0. The zero "cancels out" any effect from the changing x-value.

Q: How can I tell this is a constant function just by looking at it?

Look for multiplication by zero in the function rule. When you see $ f(x) = x \times 0 $ or any expression times 0, the result is always the same constant value.

Q: What does the graph of this function look like?

It's a horizontal line at y = 0 (the x-axis). Every point has coordinates (x, 0) where x can be any number, but y is always 0.

Q: Is this different from f(x) = 0?

No difference at all! Both $ f(x) = x \times 0 $ and $ f(x) = 0 $ give the same constant function. The multiplication by zero simplifies to just 0.

Constant Functions with Zero Multiplication

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.

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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?

00:05 Let's draw the function graph

00:12 Let's substitute X values and find the corresponding Y values

00:36 We can see that Y values are always equal, therefore the function is constant

00:42 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 with a graph or table.

For each number, multiply by 0.

Step-by-step solution

The function is:

$f(x)=x\times0$

Let's start by assuming that x equals 0:

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

Now let's assume that x equals 1:

$f(1)=1\times0=0$

Now let's assume that x equals -1:

$f(-1)=(-1)\times0=0$

Now let's assume that x equals 2:

$f(2)=2\times0=0$

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

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

Final Answer

Constant

Key Points to Remember

Essential concepts to master this topic

Zero Property: Any number multiplied by zero equals zero
Function Form: $f(x) = x \times 0 = 0$ for all x-values
Graph Check: Plot points like (-1,0), (0,0), (1,0) - all lie on y = 0 ✓

Common Mistakes

Avoid these frequent errors

Confusing zero multiplication with variable behavior
Don't think f(x) = x × 0 changes as x changes = wrong conclusion about increasing/decreasing! Zero times anything is always zero, so the output never changes. Always remember that multiplication by zero creates a constant function.

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 doesn't the function increase when x gets bigger?

Because anything times zero equals zero! Even if x = 1000, we get 1000 × 0 = 0. The zero "cancels out" any effect from the changing x-value.

How can I tell this is a constant function just by looking at it?

Look for multiplication by zero in the function rule. When you see $f(x) = x \times 0$ or any expression times 0, the result is always the same constant value.

What does the graph of this function look like?

It's a horizontal line at y = 0 (the x-axis). Every point has coordinates (x, 0) where x can be any number, but y is always 0.

Is this different from f(x) = 0?

No difference at all! Both $f(x) = x \times 0$ and $f(x) = 0$ give the same constant function. The multiplication by zero simplifies to just 0.

Can a function be both increasing and decreasing?

No! A function can be increasing in some intervals and decreasing in others, but constant functions like this one are neither increasing nor decreasing - they stay the same everywhere.

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