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## Answered question

2021-08-12

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

### Answer & Explanation

toroztatG

Skilled2021-08-13Added 98 answers

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

Expert2023-06-14Added 656 answers

To solve the problem, let's consider a function $f\left(x\right)$ 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 $f\left(x\right)$ is one-to-one, it means that for every ${x}_{1}$ and ${x}_{2}$ in the domain of $f\left(x\right)$, if ${x}_{1}\ne {x}_{2}$, then $f\left({x}_{1}\right)\ne f\left({x}_{2}\right)$. In other words, the function assigns a unique value to each input.
Now, let's focus on the horizontal line intersecting the graph of $f\left(x\right)$. 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 $f\left(x\right)=c$. This means we are looking for values of $x$ that satisfy $f\left(x\right)=c$.
If $f\left(x\right)=c$ has a solution, it means that there exists an $x$ such that $f\left(x\right)$ is equal to the constant $c$. Since $f\left(x\right)$ 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

Skilled2023-06-14Added 184 answers

The number of intersections of a horizontal line with the graph of a one-to-one function can be represented by the equation:
$\overline{)n=0\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}n=1}$ 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

Expert2023-06-14Added 669 answers

Step 1:
Let $f\left(x\right)$ be a one-to-one function. By definition, this means that for every ${x}_{1}$ and ${x}_{2}$ in the domain of the function, if $f\left({x}_{1}\right)=f\left({x}_{2}\right)$, then ${x}_{1}={x}_{2}$. In other words, no two distinct inputs can yield the same output.
Now, let's analyze the intersection of the graph of $f\left(x\right)$ with a horizontal line at a specific value, $c$. This intersection occurs when there exists an $x$ such that $f\left(x\right)=c$.
To find the number of intersections, we can set up the equation $f\left(x\right)=c$ and solve for $x$. The number of solutions will give us the number of times the horizontal line intersects the graph.
Let's express this equation using LaTeX markup:
$f\left(x\right)=c$
Step 2:
Now, depending on the specific function $f\left(x\right)$, 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 $f\left(x\right)$.
- 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.

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