Function Behavior Analysis: Multiplying Numbers by -1

Q: Why is f(x) = -x decreasing when f(-1) = 1 is positive?

Great question! Look at the overall pattern, not individual outputs. As x goes from -1 to 0 to 1 to 2, the outputs go from 1 to 0 to -1 to -2. The y-values are getting smaller - that's decreasing!

Q: How can I quickly tell if a linear function is increasing or decreasing?

Just look at the coefficient of x! If it's positive, the function increases. If it's negative (like -1 in our case), the function decreases. It's that simple!

Q: What does 'multiply by (-1)' mean in the problem?

It means f(x) = -1 × x = -x. For any input number, you multiply it by -1 to get the output. So f(3) = -1 × 3 = -3.

Q: Do I need to test many points to determine if it's decreasing?

No! For linear functions, testing just 2-3 points is enough. If each larger x-value gives a smaller y-value, it's decreasing. The graph confirms this pattern continues everywhere.

Q: Can a function be both increasing and decreasing?

Not a linear function! Linear functions have constant slope, so they're either always increasing, always decreasing, or constant. Only curved functions can increase in some parts and decrease in others.

Linear Function Behavior with Negative Slope

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 $(-1)$ .

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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?

00:03 Let's draw the function graph

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

00:45 We can see that the function is decreasing

00:49 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 $(-1)$ .

Step-by-step solution

The function is:

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

Let's start by assuming that x equals 0:

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

Now let's assume that x equals minus 1:

$f(-1)=(-1)\times(-1)=1$

Now let's assume that x equals 1:

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

Now let's assume that x equals 2:

$f(2)=(-1)\times2=-2$

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

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

Final Answer

Decreasing

Key Points to Remember

Essential concepts to master this topic

Slope Rule: Negative coefficient means function decreases as x increases
Test Method: Calculate $f(1) = -1$ and $f(2) = -2$
Graph Check: Plot points and verify line slopes downward from left to right ✓

Common Mistakes

Avoid these frequent errors

Confusing negative coefficient with increasing function
Don't think f(x) = -x is increasing because negative times negative gives positive! This ignores that as x increases, -x becomes more negative (decreases). Always remember: negative coefficient = decreasing function, regardless of sign of input values.

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 is f(x) = -x decreasing when f(-1) = 1 is positive?

Great question! Look at the overall pattern, not individual outputs. As x goes from -1 to 0 to 1 to 2, the outputs go from 1 to 0 to -1 to -2. The y-values are getting smaller - that's decreasing!

How can I quickly tell if a linear function is increasing or decreasing?

Just look at the coefficient of x! If it's positive, the function increases. If it's negative (like -1 in our case), the function decreases. It's that simple!

What does 'multiply by (-1)' mean in the problem?

It means f(x) = -1 × x = -x. For any input number, you multiply it by -1 to get the output. So f(3) = -1 × 3 = -3.

Do I need to test many points to determine if it's decreasing?

No! For linear functions, testing just 2-3 points is enough. If each larger x-value gives a smaller y-value, it's decreasing. The graph confirms this pattern continues everywhere.

Can a function be both increasing and decreasing?

Not a linear function! Linear functions have constant slope, so they're either always increasing, always decreasing, or constant. Only curved functions can increase in some parts and decrease in others.

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