Increasing Function Intervals: Graph-Based Domain Investigation

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

Imagine walking along the curve from left to right. If you're walking uphill (the curve is rising), the function is increasing. If you're walking downhill (the curve is falling), it's decreasing.

Q: What does the notation -1 < x < 1 mean?

This means x is between -1 and 1, but not equal to -1 or 1. It's the same as writing $ 1 > x > -1 $ - just the inequality signs are flipped around!

Q: How do I read the domain from a graph?

Look at the x-axis (horizontal axis) and identify the interval where the behavior you're asked about occurs. The domain is the set of all possible x-values for that specific behavior.

Function Behavior with Graph Analysis

In what domain does the function increase?

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Step-by-step video solution

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00:00 Find the increasing domain of the function

00:04 The function increases when X and Y values increase simultaneously

00:07 This is the increasing domain of the function

00:11 Find the start and end points of the increasing domain

00:16 And this is the solution to the question

Step-by-step written solution

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Understand the problem

In what domain does the function increase?

Step-by-step solution

First we need to remember that the function increases if the X and Y values increase simultaneously.

Conversely, 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 $1 > x > -1$ , meaning the Y values are increasing.

Final Answer

$1 > x > -1$

Key Points to Remember

Essential concepts to master this topic

Increasing Rule: Function rises when both x and y values increase together
Graph Reading: Follow curve from left to right, noting where y-values go up
Domain Check: Verify interval notation matches graph sections where function rises ✓

Common Mistakes

Avoid these frequent errors

Confusing increasing and decreasing intervals
Don't look at where the graph goes down and call it increasing! This gives the opposite interval. The function decreases when y-values drop as x increases. Always identify where the curve climbs upward 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 do I tell if a function is increasing just by looking at the graph?

Imagine walking along the curve from left to right. If you're walking uphill (the curve is rising), the function is increasing. If you're walking downhill (the curve is falling), it's decreasing.

What does the notation -1 < x < 1 mean?

This means x is between -1 and 1, but not equal to -1 or 1. It's the same as writing $1 > x > -1$ - just the inequality signs are flipped around!

Why doesn't the function increase everywhere on the graph?

Functions can have different behaviors in different sections! This graph shows the function increasing between x = -1 and x = 1, but decreasing outside that interval.

How do I read the domain from a graph?

Look at the x-axis (horizontal axis) and identify the interval where the behavior you're asked about occurs. The domain is the set of all possible x-values for that specific behavior.

What if the graph touches but doesn't cross x = -1 or x = 1?

Great observation! If the function changes from increasing to decreasing (or vice versa) exactly at those points, we typically exclude the boundary points from the increasing interval.

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