# 微积分网课代修|预备微积分代写precalculus辅导|MATH1730 Using the Vertical Line Test

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## 微积分网课代修|预备微积分代写precalculus辅导|Using the Vertical Line Test

As we have seen in some examples above, we can represent a function using a graph. Graphs display a great many input-output pairs in a small space. The visual information they provide often makes relationships easier to understand. By convention, graphs are typically constructed with the input values along the horizontal axis and the output values along the vertical axis.

The most common graphs name the input value $x$ and the output value $y$, and we say $y$ is a function of $x$, or $y=f(x)$ when the function is named $f$. The graph of the function is the set of all points $(x, y)$ in the plane that satisfies the equation $y=f(x)$. If the function is defined for only a few input values, then the graph of the function is only a few points, where the $x$-coordinate of each point is an input value and the $y$-coordinate of each point is the corresponding output value. For example, the black dots on the graph in Figure 10 tell us that $f(0)=2$ and $f(6)=1$. However, the set of all points $(x, y)$ satisfying $y=f(x)$ is a curve. The curve shown includes $(0,2)$ and $(6,1)$ because the curve passes through those points.

The vertical line test can be used to determine whether a graph represents a function. If we can draw any vertical line that intersects a graph more than once, then the graph does not define a function because a function has only one output value for each input value. See Figure 11 .

## 微积分网课代修|预备微积分代写precalculus辅导|Using the Horizontal Line Test

Once we have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to use the horizontal line test. Draw horizontal lines through the graph. If any horizontal line intersects the graph more than once, then the graph does not represent a one-to-one function.
How To…
Given a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one function.

1. Inspect the graph to see if any horizontal line drawn would intersect the curve more than once.
2. If there is any such line, determine that the function is not one-to-one.Example 15 Horizontal Line Test
Consider the functions shown in Figure 12(a) and Figure 12(b). Are either of the functions one-to-one?
Solution The function in Figure 12(a) is not one-to-one. The horizontal line shown in Figure 15 intersects the graph of the function at two points (and we can even find horizontal lines that intersect it at three points.)

## 微积分网课代修|预备微积分代写precalculus辅导|Using the Horizontal Line Test

1. 检查图形以查看绘制的任何水平线是否会与曲线多次相交。
2. 如果有任何这样的线，则确定该函数不是一对一的。示例 15 水平线测试
考虑图 12(a) 和图 12(b) 中所示的函数。任何一个功能都是一对一的吗？
解决方案 图 12(a) 中的函数不是一对一的。图 15 所示的水平线在两点与函数图相交（我们甚至可以找到在三点与它相交的水平线。）