Function Analysis: Determining Behavior When Dividing by -1

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

Look at the coefficient of x! If it's positive, the function increases. If it's negative (like $ \frac{x}{-1} = -x $), the function decreases.

Q: What points should I test to check my answer?

Pick at least three simple values like x = -1, 0, and 1. Calculate f(x) for each, then see if the y-values increase or decrease as x increases.

Q: Is f(x) = x/(-1) the same as f(x) = -x?

Yes! $ \frac{x}{-1} = -x $ because dividing by -1 is the same as multiplying by -1. Both represent the same decreasing linear function.

Q: Can a function be both increasing and decreasing?

Not for linear functions! A linear function has the same rate of change everywhere, so it's either always increasing, always decreasing, or constant. Only curved functions can change direction.

Function Monotonicity with Negative Coefficients

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 $(-1)$ .

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:11 Is the function increasing, decreasing, or staying constant?

00:16 First, let's draw the graph of the function.

00:22 We'll substitute some X values to find their matching Y values.

00:39 Now, let's try some negative X values to see the graph's left side.

00:46 We observe that the graph is always decreasing.

00:50 And that's how we solve this 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:

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

Step-by-step solution

The function is:

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

Let's start by assuming that x equals 0:

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

Now let's assume that x equals 1:

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

Now let's assume that x equals 2:

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

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

We see that we got a decreasing function.

Final Answer

Decreasing

Key Points to Remember

Essential concepts to master this topic

Rule: Dividing by negative numbers reverses the order relationship
Technique: Calculate f(x) = x/(-1) = -x for test points
Check: Compare function values: as x increases, f(x) decreases ✓

Common Mistakes

Avoid these frequent errors

Ignoring the negative sign when determining function behavior
Don't treat $f(x) = \frac{x}{-1}$ like $f(x) = x$ = wrong conclusion about increasing! The negative coefficient flips the relationship completely. Always remember that multiplying or dividing by negative values reverses the direction of change.

Practice Quiz

Test your knowledge with interactive questions

Does the function in the graph decrease throughout?

FAQ

Everything you need to know about this question

Why does dividing by -1 make the function decreasing?

When you divide by -1, you're essentially multiplying by -1, which flips the sign of every input. So as x gets bigger (more positive), $f(x) = -x$ gets smaller (more negative).

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

Look at the coefficient of x! If it's positive, the function increases. If it's negative (like $\frac{x}{-1} = -x$ ), the function decreases.

What points should I test to check my answer?

Pick at least three simple values like x = -1, 0, and 1. Calculate f(x) for each, then see if the y-values increase or decrease as x increases.

Is f(x) = x/(-1) the same as f(x) = -x?

Yes! $\frac{x}{-1} = -x$ because dividing by -1 is the same as multiplying by -1. Both represent the same decreasing linear function.

Can a function be both increasing and decreasing?

Not for linear functions! A linear function has the same rate of change everywhere, so it's either always increasing, always decreasing, or constant. Only curved functions can change direction.

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