Function Domain Analysis: Finding Intervals of Increase with x=1.1

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

Look for sections where the graph goes upward from left to right. If you imagine walking along the curve from left to right and you're going uphill, the function is increasing in that interval.

Q: What's the difference between 1.1 > x > 0 and 0 < x < 1.1?

They mean exactly the same thing! Both represent all numbers between 0 and 1.1. The notation 0 smaller to larger values.

Q: What does the black vertical line at x = 1.1 represent?

The black line marks the boundary point where the function's behavior changes. It helps you identify exactly where the function stops increasing and starts decreasing.

Q: Can I include the endpoints 0 and 1.1 in my answer?

Based on the graph, we use strict inequalities () rather than ≤ or ≥. The function appears to have turning points at these values, so they're typically not included in the increasing interval.

Function Behavior with Graphical Interpretation

In what domain does the function increase?

Black line: $x=1.1$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the increasing domain of the function

00:03 The function increases when X and Y values increase simultaneously

00:07 Let's find the increasing domain of the function

00:11 We'll precisely identify the starting and ending points of the increase

00:16 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?

Black line: $x=1.1$

Step-by-step solution

Remember that a function is increasing if the $x$ values and $y$ values are increasing simultaneously.

A function is decreasing if the X values are increasing and the Y values are decreasing simultaneously.

In the plotted graph, we can see that in the domain $1.1 > x > 0$ the function is increasing—meaning the $y$ values are increasing.

Final Answer

$1.1 > x > 0$

Key Points to Remember

Essential concepts to master this topic

Increasing Function: When x increases, y also increases (upward trend)
Visual Check: Look for graph segments sloping upward from left to right
Domain Reading: Read x-values from left boundary to right boundary ✓

Common Mistakes

Avoid these frequent errors

Confusing inequality direction when reading domains
Don't write 1.1 < x < 0 which is mathematically impossible! This creates an empty set since no number can be both less than 1.1 AND less than 0 simultaneously. Always write domain intervals with smaller value first: 0 < x < 1.1.

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

Look for sections where the graph goes upward from left to right. If you imagine walking along the curve from left to right and you're going uphill, the function is increasing in that interval.

What's the difference between 1.1 > x > 0 and 0 < x < 1.1?

They mean exactly the same thing! Both represent all numbers between 0 and 1.1. The notation 0 < x < 1.1 is more standard because we read from smaller to larger values.

Why isn't the answer x > 0 if the function increases after x = 0?

Look carefully at the graph! The function increases from x = 0 to x = 1.1, but then it starts decreasing after x = 1.1. So x > 0 includes values where the function is decreasing.

What does the black vertical line at x = 1.1 represent?

The black line marks the boundary point where the function's behavior changes. It helps you identify exactly where the function stops increasing and starts decreasing.

Can I include the endpoints 0 and 1.1 in my answer?

Based on the graph, we use strict inequalities (< and >) rather than ≤ or ≥. The function appears to have turning points at these values, so they're typically not included in the increasing interval.

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