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HaLAA
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Homework Statement
A one to one function f: ℝ→ℝ is monotone, True or False
Homework Equations
The Attempt at a Solution
I think the statement is false, for example: Let I =[0,1]∪[2,3] f(x)=x if x∈[0,1], f(x)=5-x,x∈[2,3]
Let f(x)=x, x is rational, f(x)=-x,x is irrational, the function is one to one,but it is jumping. Does this example apply?andrewkirk said:You have not defined the value of f(x) outside of I, so the example does not meet the requirements of the question, which include that the domain be all of ##\mathbb{R}##.
I think the statement is false, but a little more work is needed to produce a counterexample.
Since both are dense, it's just a big X. And it leads to interesting philosophical questions: what one draws is always discrete for you put carbon atoms on the paper. (I apologize, if that remark should be regarded as improper.)andrewkirk said:(albeit harder to visualize).
Of course you could have f(x) = x outside of the interval (-1, 1) and f(x) = -x on the interval (-1, 1) .HaLAA said:Let f(x)=x, x is rational, f(x)=-x,x is irrational, the function is one to one,but it is jumping. Does this example apply?
A one-to-one function is a type of mathematical function where each input value has a unique output value. In other words, for every x-value, there is only one corresponding y-value. This means that no two different input values can produce the same output value.
A monotone function is a type of mathematical function where the output values either consistently increase or decrease as the input values increase. In other words, the function either has a consistently positive or negative slope.
A function is one-to-one if and only if each input value has a unique output value. This can be determined by using the vertical line test. This test involves drawing a vertical line through the graph of the function. If the line intersects the graph at only one point, then the function is one-to-one. If the line intersects the graph at more than one point, then the function is not one-to-one.
Yes, it is possible for a function to be both one-to-one and monotone. This would mean that the function has a unique output value for each input value and the output values consistently increase or decrease as the input values increase.
A one-to-one monotone function is significant because it guarantees that the function has an inverse. This means that the function can be "reversed" to find the original input value from a given output value. This is important in many real-world applications, such as finding the original cost of a discounted item or the initial population of a species based on its current population.