Function Domain Analysis: Finding Intervals of Increase from Graph

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

Look at the slope direction! If the graph goes upward as you move from left to right, the function is increasing. If it goes downward, it's decreasing.

Q: What's the difference between where y > 0 and where the function increases?

y > 0 tells you where the graph is above the x-axis. Increasing means the graph slopes upward. These are completely different concepts!

Q: What if the graph has both increasing and decreasing parts?

Then you identify multiple intervals! List each separate interval where the function increases, like: x 3.

Q: How do I write my answer for increasing intervals?

Use inequality notation like $ x > 0 $, or interval notation like (0, ∞). Both show the same domain where the function increases.

Function Behavior with Graphical Analysis

In what domain does the function increase?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the domain where the function increases

00:04 A function increases if X and Y values increase simultaneously

00:07 We can see that while X is less than 0, the function decreases

00:13 We notice there's a change at the origin point

00:17 The function increases as long as X is greater than 0

00:20 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

In what domain does the function increase?

Step-by-step solution

Let's remember that the function increases if the $x$ values and $y$ values increase simultaneously.

On the other hand, the function decreases if the $x$ values increase while the $y$ values decrease simultaneously.

In the given graph, we can see that the function increases in the domain where $x > 0$ ; in other words, where the $y$ values are increasing.

Final Answer

$x > 0$

Key Points to Remember

Essential concepts to master this topic

Increasing Functions: y-values rise as x-values move right on graph
Visual Check: Function increases where graph slopes upward from left to right
Verification: Pick two points: if x₂ > x₁ and y₂ > y₁, function increases ✓

Common Mistakes

Avoid these frequent errors

Confusing x and y coordinates when determining increase
Don't look at where y > 0 to find increasing intervals = wrong domain! This confuses positive y-values with increasing behavior. Always check if y-values rise as x-values increase from left to right.

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

Look at the slope direction! If the graph goes upward as you move from left to right, the function is increasing. If it goes downward, it's decreasing.

What's the difference between where y > 0 and where the function increases?

y > 0 tells you where the graph is above the x-axis. Increasing means the graph slopes upward. These are completely different concepts!

Can a function increase in the negative x region?

Absolutely! A function can increase anywhere on the graph. In this problem, the function increases for $x > 0$ , but functions can increase for negative x-values too.

What if the graph has both increasing and decreasing parts?

Then you identify multiple intervals! List each separate interval where the function increases, like: x < -2 and x > 3.

How do I write my answer for increasing intervals?

Use inequality notation like $x > 0$ , or interval notation like (0, ∞). Both show the same domain where the function increases.

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