- #1
DecayProduct
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Why does the vertical line test work? If the line hits the graph more than once, it is said to not be a function. But why?
Non-functions and the vertical line test
In summary: Yes, I think that is exactly what it is for. If a function is one-to-one, then every point on the graph corresponds to a unique value of the output (ie. if x is the input, then y=f(x)) Otherwise, the function is not one-to-one and there may be multiple values that correspond to a single input.
- #2
tiny-tim
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DecayProduct said:
Hi DecayProduct !
Because a map is a function from one space to another …
so, for any point x in the first space, there must be an f(x) in the second space.
There can't be more than one f(x) for the same x.
And if the (vertical) line hit the graph more than once, then that would make two f(x)'s for one x!
- #3
HallsofIvy
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The definition of a function is "a set of ordered pairs (x, f(x)) so that now two pairs have the same first member" or, equivalently, a mapping x-> f(x) so that the same x does not give two different values for f(x).
The graph of a relation consists of the points (x,f(x)). A vertical line corresponds to a specific value of x. If a vertical line crosses the graph at two points, that means we have two different values of y associated with the same value of x: f(x)= y1 and f(y)= y2 for different y1 and y2. That violates the definition of "function".
The graph of a relation consists of the points (x,f(x)). A vertical line corresponds to a specific value of x. If a vertical line crosses the graph at two points, that means we have two different values of y associated with the same value of x: f(x)= y1 and f(y)= y2 for different y1 and y2. That violates the definition of "function".
- #4
DecayProduct
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Thanks to both of you! That makes sense. I theorized that, but not so eloquently, and of course, wasn't sure if that was even close to why.
- #5
roam
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A curve in the xy plane is the graph of some function ƒ IF no vertical line intersects the curve more than once.
You're right, but don't confuse the "vertical line test" with the "horizontal line test", this is obviously one of the mistakes I made when I started calculus.
The horizontal line test tells you wether a function is one to one and onto or not (ie. a horizontal line can't intersect a bijective function "1:1" more than once)
Be careful with that!
You're right, but don't confuse the "vertical line test" with the "horizontal line test", this is obviously one of the mistakes I made when I started calculus.
The horizontal line test tells you wether a function is one to one and onto or not (ie. a horizontal line can't intersect a bijective function "1:1" more than once)
Be careful with that!
FAQ: Non-functions and the vertical line test
1. What is a non-function?
A non-function is a relation in which one input value is associated with more than one output value. In other words, there are multiple y-values for a single x-value, making it difficult to graph and analyze.
2. How can I identify a non-function?
A non-function can be identified by using the vertical line test. This test involves drawing a vertical line on a graph and observing if it intersects the graph more than once. If it does, then the relation is a non-function.
3. Why is the vertical line test important?
The vertical line test is important because it helps us determine if a relation is a function or not. Functions have a unique output for every input, making them easier to graph and analyze. Additionally, the vertical line test is a visual representation of the definition of a function.
4. Can a non-function be written as a mathematical equation?
Yes, a non-function can be written as a mathematical equation. However, the equation will not pass the vertical line test, indicating that it is not a function. This can be seen in equations with multiple y-values for a single x-value, such as y = ±√x.
5. Are there any real-life examples of non-functions?
Yes, there are many real-life examples of non-functions. One example is a rollercoaster ride, where the height of the rider at a certain time can have multiple speed values. Another example is a person's weight over time, as their weight may fluctuate even if their age stays the same.