SOLUTION: A horizontal line intersects a function at two points. Is the function invertible? My friend says the answer is no. Please explain why?

Question 895430: A horizontal line intersects a function at two points. Is the function invertible?
My friend says the answer is no. Please explain why?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
the answer appears to be yes.
if the original function doesn't pass the horizontal line test then the inverse function will not pass the vertical line test.

note that the inverse equation of y = f(x) is x = f(y)

for example, the inverse equation of y = x^2 is x = y^2

you can replace x with y and solve for y to get y = plus or minus sqrt(x) if you like, but the inverse function starts off as x = y^2 and can be graphed as x = y^2.

you can replace y with x and x with y in that equation and then solve for y because it's doable.

in more complicated equations, it's not always possible.

for example:

y = x^3 + 5x^2 has an inverse function of x = y^3 + 5y^2

how do you replace x with y and y with x and then solve for y in that equation?

it's not so easy.

you get x = y^3 + 5y^2 and you are asked to solve for y.

i don't know how to do it.

but i do know that the inverse equation is x = y^3 + 5y^2 and i can graph that if i care to.

the following graphs will show you some functions and their inverse equations and show you that, if the original equation does not pass the horizontal line test, then the inverse equation will not pass the vertical line test.

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