Function Domain Analysis: Finding Intervals Where f(x) is Increasing

Q: How do I know when a function is increasing from a table?

A function is increasing when both x-values and y-values go up together. Look at consecutive points in the table - if y gets bigger as x gets bigger, it's increasing in that interval.

Q: Why isn't the answer x < 2.5 since the function increases until x = 2.5?

Because we need to find where it's consistently increasing. From the table, y decreases before x = 0 and after x = 2.5. Only between \( 0 does it truly increase.

Q: What's the difference between increasing and decreasing?

Increasing: As x goes up, y goes up too (positive slope)Decreasing: As x goes up, y goes down (negative slope)

Q: Can a function be increasing in multiple separate intervals?

Yes! A function can increase in some intervals and decrease in others. That's why we need to specify the exact interval, like \( 0 , not just say 'the function increases.'

Q: How do I read the interval notation 2.5 > x > 0?

This means 'x is between 0 and 2.5' - x is greater than 0 AND less than 2.5. You can also write it as \( 0 which means the same thing.

Function Behavior with Table Analysis

In which domain is the function increasing?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:03 Let's find where the function is increasing.

00:07 The function increases when both X and Y go up together.

00:13 From this point on, the function starts to rise.

00:19 Then here, the function begins to fall again.

00:25 Let's figure out where the function is increasing.

00:36 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

In which domain is the function increasing?

Step-by-step solution

Note that the function increases when X values and Y values increase simultaneously.

On the other hand, the function decreases when X values increase and Y values decrease simultaneously.

According to the given value table, we can see that in the domain where $2.5 > x > 0$ the X values and Y values increase simultaneously.

Therefore, the function increases in the domain where

$2.5 > x > 0$

Final Answer

$2.5>x>0$

Key Points to Remember

Essential concepts to master this topic

Increasing Function: Both x-values and y-values grow together
Technique: Compare consecutive points: from x=0 to x=2.5, y goes 0→2
Check: Verify all intervals: only 0 < x < 2.5 shows consistent increase ✓

Common Mistakes

Avoid these frequent errors

Looking at entire domain instead of specific intervals
Don't say the function increases everywhere because some y-values are positive = missing the decreasing parts! The function can increase in some intervals and decrease in others. Always examine each interval separately by comparing consecutive y-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

How do I know when a function is increasing from a table?

A function is increasing when both x-values and y-values go up together. Look at consecutive points in the table - if y gets bigger as x gets bigger, it's increasing in that interval.

Why isn't the answer x < 2.5 since the function increases until x = 2.5?

Because we need to find where it's consistently increasing. From the table, y decreases before x = 0 and after x = 2.5. Only between $0 < x < 2.5$ does it truly increase.

What's the difference between increasing and decreasing?

Increasing: As x goes up, y goes up too (positive slope)
Decreasing: As x goes up, y goes down (negative slope)

Can a function be increasing in multiple separate intervals?

Yes! A function can increase in some intervals and decrease in others. That's why we need to specify the exact interval, like $0 < x < 2.5$ , not just say 'the function increases.'

How do I read the interval notation 2.5 > x > 0?

This means 'x is between 0 and 2.5' - x is greater than 0 AND less than 2.5. You can also write it as $0 < x < 2.5$ which means the same thing.

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