Question

How do I know if an equation is a one-to-one function?

Answer 1

For every y which has been assigned to the x's no other x has that y

Answer 2

for every y there is one and only one value x

Answer 3

That first sentence can get confusing O_O

Answer 4

a visual way to check

if the graph passes both the vertical and horizontal line tests, then you have a one-to-one function

Answer 5

Yeah it kind of is. I know the vertical and horizontal line tests but for example if I have f(x)-x^3-8 how would I know if it was a one-to-one or not? I solved for the inverse function which is Cube root of 3 x plus 8.. Does that make sense? lol

Answer 6

If f(x) was one-to-one, then

f(a) = f(b) if and only if a = b

Answer 7

so start with f(a) = f(b) and if we can prove that a = b, then f(x) is one to one


f(x) = x^3 - 8

f(a) = a^3 - 8

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f(x) = x^3 - 8

f(b) = b^3 - 8

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Now we start with f(a) = f(b) and prove that a = b

f(a) = f(b)

a^3 - 8 = b^3 - 8

a^3 - 8 + 8 = b^3 - 8 + 8

a^3 + 0 =b^3 + 0

a^3 = b^3

a = b ... take the cube root of both sides

Answer 8

we could easily go in reverse to show that if a = b, then f(a) = f(b)

therefore, f(x) = x^3 - 8 is indeed one-to-one

Answer 9

Thank you so much! That was very helpful!

Answer 10

you're welcome

Answer 11

Thanks for the example Jim, helped me out as well actually :)

Answer 12

glad to be of help to you both