How many times can a horizontal line intersect the graph of a function that is one-to-one?

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toroztatG · Accepted Answer

Suppose f is an one-to-one function.
If (a,b) lies in the graph of f, then there does not exists any x with x!=a such that (x,b) also lies in the graph of f (because f is one-to-one).
This means if f is one-to-one, then a horizontal line intersect the graph of f at most once.
Thus a horizontal line intersect the graph of an one-to-one function at most once.

RizerMix · Accepted Answer

To solve the problem, let&#039;s consider a function f(x) that is one-to-one. In order to find how many times a horizontal line intersects the graph of this function, we need to analyze the behavior of the function.Since f(x) is one-to-one, it means that for every x1 and x2 in the domain of f(x), if x1≠x2, then f(x1)≠f(x2). In other words, the function assigns a unique value to each input.Now, let&#039;s focus on the horizontal line intersecting the graph of f(x). We can represent this line as y=c, where c is a constant.To find the points of intersection, we need to solve the equation f(x)=c. This means we are looking for values of x that satisfy f(x)=c.If f(x)=c has a solution, it means that there exists an x such that f(x) is equal to the constant c. Since f(x) is one-to-one, there can be at most one solution for x.Therefore, a horizontal line can intersect the graph of a one-to-one function at most once.

Don Sumner · Accepted Answer

The number of intersections of a horizontal line with the graph of a one-to-one function can be represented by the equation:

n = 0 or n = 1

where

n

represents the number of intersections.
Note: that in this context,

n

can either be 0 or 1, indicating that a horizontal line can intersect the graph of a one-to-one function either zero times or exactly once.

Vasquez · Accepted Answer

Step 1:Let f(x) be a one-to-one function. By definition, this means that for every x1 and x2 in the domain of the function, if f(x1)=f(x2), then x1=x2. In other words, no two distinct inputs can yield the same output.Now, let&#039;s analyze the intersection of the graph of f(x) with a horizontal line at a specific value, c. This intersection occurs when there exists an x such that f(x)=c.To find the number of intersections, we can set up the equation f(x)=c and solve for x. The number of solutions will give us the number of times the horizontal line intersects the graph.Let&#039;s express this equation using LaTeX markup:f(x)=cStep 2:Now, depending on the specific function f(x), the number of solutions to this equation can vary. It is possible to have zero, one, or multiple solutions.- If there are no solutions, the horizontal line does not intersect the graph of f(x).- If there is one solution, the horizontal line intersects the graph at a single point.- If there are multiple solutions, the horizontal line intersects the graph at multiple points.In conclusion, the number of times a horizontal line can intersect the graph of a one-to-one function can be zero, one, or multiple times, depending on the specific function and the value of the horizontal line.

How many times can a horizontal line intersect the graph

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