Vertical Line Test/Horizontal Line Test

[nycmathguy]

What is the most basic difference between a vertical line test and a horizontal line test? I think if a graph fails the horizontal line test, the function is not one-to-one. Yes? If a graph fails the vertical line test, there is no function. Yes?

 

[MathLover1]

So in short, if you have a curve, the vertical line test checks if that curve is a function, and the horizontal line test checks whether the inverse of that curve is a function.

 

[nycmathguy]

So in short, if you have a curve, the vertical line test checks if that curve is a function, and the horizontal line test checks whether the inverse of that curve is a function.

I understand. However, can you provide one graph for each case for study purposes?

 

[MathLover1]

the vertical line test: function y=x+1

=> y=x+1 is a function

or y=x^2-2

=> y=x^2-2 is a function


or, graph of the circle x^2 + y^2 = 9

=>x^2 + y^2 = 9 is NOT a function


the horizontal line test:

graph of the line y=-x+2
the inverse:
y=-x+2=>swap variables
x=-y+2..........solve for y
y=-x+2 (same as original function)

=> is one-to-one function

Graph of the rational function
f(x)= 1/(x+1)

inverse:
y= 1/(x+1)
x= 1/(y+1)
(y+1)= 1/x
y= 1/x-1

=> is one-to-one function

 

[nycmathguy]

the vertical line test: function y=x+1

=> y=x+1 is a function

or y=x^2-2

=> y=x^2-2 is a function


or, graph of the circle x^2 + y^2 = 9

=>x^2 + y^2 = 9 is NOT a function


the horizontal line test:

graph of the line y=-x+2
the inverse:
y=-x+2=>swap variables
x=-y+2..........solve for y
y=-x+2 (same as original function)

=> is one-to-one function

Graph of the rational function
f(x)= 1/(x+1)

inverse:
y= 1/(x+1)
x= 1/(y+1)
(y+1)= 1/x
y= 1/x-1

=> is one-to-one function

Beautifully done.