Proving One-to-One Function Strictly Increasing on Interval I

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In summary, a one-to-one function is a type of mathematical function in which each input has a unique output. A function is strictly increasing if its output values increase as its input values increase. To prove that a function is one-to-one, you can use the horizontal line test, and to show that a function is strictly increasing on a given interval, you can use the first derivative test. Additionally, it is possible for a function to be both one-to-one and strictly increasing, which is useful in various mathematical applications.
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I am having trouble with this study question for my final:

A function from the real numbers to the real numbers is one to one on an interval I if it is strictly increasing on that interval.

I am not quite sure how to prove it, I know that the use of strictly increasing is important as far as if x1 < x2, then f(x1) < f(x2). A hint I was told to use is contradiction.

Any help please!
 
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FAQ: Proving One-to-One Function Strictly Increasing on Interval I

What is a one-to-one function?

A one-to-one function is a type of mathematical function in which each input has a unique output. This means that no two different inputs can have the same output. In other words, each input is mapped to a unique output.

What does it mean for a function to be strictly increasing?

A function is strictly increasing if its output values increase as its input values increase. In other words, as the input increases, the output also increases without any decrease in between. This is also known as a positive slope or a function that is always going up.

How can I prove that a function is one-to-one?

To prove that a function is one-to-one, you can use the horizontal line test. This involves drawing horizontal lines across the graph of the function. If the line intersects the graph at more than one point, then the function is not one-to-one. If the line only intersects the graph at one point, then the function is one-to-one.

How do I show that a function is strictly increasing on an interval?

To show that a function is strictly increasing on a given interval, you can use the first derivative test. This involves finding the first derivative of the function and evaluating it at points within the interval. If the derivative is positive at all points, then the function is strictly increasing on that interval.

Can a function be both one-to-one and strictly increasing?

Yes, a function can be both one-to-one and strictly increasing. This means that each input has a unique output and that the output values increase as the input values increase. This type of function is useful in many areas of mathematics, such as in modeling real-world situations and in solving equations.

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