Proving f(x) intersects a graph when given an interval

In summary, the conversation discusses how to prove that the graph of a function must intersect a specific line. The solution involves looking at the difference between the function and the line and using the Intermediate Value Theorem. It is also noted that this proof only applies to continuous functions.
  • #1
KataKoniK
1,347
0
Hi,

Does anyone here have any pointers on how to even start the following question?

Suppose for all x element [0, 1] we have f(x) element [0, 1]. Prove that the graph f(x) must intersect the line y = 1 - x

Any help would be great thanks.
 
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  • #2
Look at the difference between f(x) and 1-x. At x = 0, the difference will either be zero or nonzero. If it is zero then you have an intersection. If the difference is not zero then it is either positive or negative but must change sign somewhere between x = 0 and x = 1 since f(0) < 1 and f(x) must reach 1 somewhere in the interval while 1-x goes to 0 at the right boundary. Since the sign of the difference changes at some point in the interval then there must be an x' for which f(x') = 1 - x'.
 
  • #3
I might point out that Tide's solution works (and, indeed, the statement is true) only if f is a continuous function on [0,1] which was not stated in the original problem:
If f(x)= 1/4 for x<= 1/2 and 3/4 for x> 1/2 then its graph does NOT intersect the line y= 1-x.
 
  • #4
Thanks for the help!
 
  • #5
i know this one!, btw its called the Intermediate value theroem
 

FAQ: Proving f(x) intersects a graph when given an interval

How do you prove that f(x) intersects a graph when given an interval?

In order to prove that f(x) intersects a graph when given an interval, you can use the Intermediate Value Theorem. This theorem states that if a function is continuous on a closed interval [a, b], and f(a) and f(b) are of opposite signs, then there exists at least one value c in the interval (a, b) such that f(c) = 0. This means that the graph of f(x) must cross the x-axis at some point between a and b, thereby intersecting the graph.

Can you use any other method to prove that f(x) intersects a graph when given an interval?

Yes, there are other methods that can be used to prove that f(x) intersects a graph when given an interval. One method is to find the x-intercepts of the function and determine if they fall within the given interval. If they do, then this proves that the graph intersects the given interval. Another method is to graph the function and visually observe if the graph crosses the x-axis within the given interval.

Is it possible for f(x) to not intersect a graph when given an interval?

Yes, it is possible for f(x) to not intersect a graph when given an interval. This can happen if the function is not continuous on the given interval or if the graph does not cross the x-axis within the given interval. Additionally, if f(a) and f(b) are both positive or both negative, the Intermediate Value Theorem cannot be applied and it is not guaranteed that the graph will intersect the x-axis within the given interval.

Can the Intermediate Value Theorem be used to prove that a graph intersects multiple points within a given interval?

Yes, the Intermediate Value Theorem can be used to prove that a graph intersects multiple points within a given interval. This is because the theorem states that there is at least one value c in the interval (a, b) such that f(c) = 0. Therefore, if there are multiple points on the graph where f(x) = 0 within the given interval, the theorem can be used to prove that the graph intersects multiple points within the interval.

Are there any limitations to using the Intermediate Value Theorem to prove that f(x) intersects a graph?

Yes, there are limitations to using the Intermediate Value Theorem to prove that f(x) intersects a graph. This theorem can only be applied to continuous functions, meaning that the function does not have any breaks or holes in its graph. Additionally, the theorem can only be applied if the function values at the endpoints of the interval are of opposite signs. If these conditions are not met, the theorem cannot be used and other methods must be used to prove the intersection of the graph within the given interval.

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