Function Analysis at -1: Determining Increasing or Decreasing Behavior

Q: How can I tell if a function is increasing just by looking at the equation?

For linear functions like $ f(x) = x - 1 $, look at the coefficient of x. Since it's +1 (positive), the function is increasing. If it were negative, the function would be decreasing.

Q: Why do we need to check multiple points?

Checking multiple points helps you see the pattern! One point only gives you a location, but comparing several points shows whether the function goes up or down as x increases.

Q: What does it mean for a function to be increasing at x = -1?

It means that around the point x = -1, as you move to the right (increasing x), the y-values also increase. For linear functions, if it's increasing at one point, it's increasing everywhere!

Q: What if the function has the same y-value for different x-values?

Then the function would be constant (neither increasing nor decreasing). This happens when the coefficient of x is zero, like $ f(x) = 5 $.

Linear Function Analysis with Increasing Behavior

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

For each number corresponding to the number $-1$ .

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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

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

00:31 Now let's substitute negative X values to check the left side of the graph

00:39 We can see that the function is increasing

00:44 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. Check your answer with a graph or table.

For each number corresponding to the number $-1$ .

Step-by-step solution

The function is:

$f(x)=x-1$

Let's start with x equal to 0:

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

Now let's assume x is equal to 1:

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

Now let's assume x is equal to 2:

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

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

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

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

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

Final Answer

Increasing

Key Points to Remember

Essential concepts to master this topic

Linear Functions: Have constant rate of change, always increasing or decreasing
Testing Method: Calculate f(-1) = -2, f(0) = -1, f(1) = 0, f(2) = 1
Verification: Plot points and check that y-values increase as x increases ✓

Common Mistakes

Avoid these frequent errors

Looking at only one point to determine behavior
Don't check just f(-1) = -2 and conclude the function is decreasing! This only tells you the value at one point, not the overall trend. Always compare multiple points or examine the slope to determine if a function is increasing or decreasing.

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

How can I tell if a function is increasing just by looking at the equation?

For linear functions like $f(x) = x - 1$ , look at the coefficient of x. Since it's +1 (positive), the function is increasing. If it were negative, the function would be decreasing.

Why do we need to check multiple points?

Checking multiple points helps you see the pattern! One point only gives you a location, but comparing several points shows whether the function goes up or down as x increases.

What does it mean for a function to be increasing at x = -1?

It means that around the point x = -1, as you move to the right (increasing x), the y-values also increase. For linear functions, if it's increasing at one point, it's increasing everywhere!

Can I use the graph instead of calculating points?

Absolutely! The graph is a great visual tool. If the line goes upward from left to right, the function is increasing. If it goes downward, it's decreasing.

What if the function has the same y-value for different x-values?

Then the function would be constant (neither increasing nor decreasing). This happens when the coefficient of x is zero, like $f(x) = 5$ .

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