Question

how would I solve and determine if function is one-to-one:
f(x)=x^2-7

Answer 1

actually, I'm not asked to solve it. I guess I couldn't not knowing what f(x) is

Answer 2

Well, it's still the Horizontal Line Test.
Since we're calling it a function, it already passed the Vertical Line Test.

Answer 3

since \(f(-1)=f(1)=-6\) it is not one to one
it is a parabola which does not pass the horizontal line test precisely because \(f(x)=f(-x)\)

Answer 4

@satellite73 how did you get f(-x)?

Answer 5

Parabolas have symmetry. With \(x^{2}\), the symmetry is across a vertical axis at x = 0. Thus GENERALLY f(x) = f(-x). Pick ANY value for 'x'.

Answer 6

so if I say x=2 then f(2)=(2)^2-7 or 4-7=-3?

Answer 7

Now do the same with x = -2. You should get the same result and you will have conclusive proof that f(x) is NOT one-to-one.

Answer 8

ok so, -2^2 is also 2 so the answer is no because it's the same?

Answer 9

Careful with that. \(-2^{2}\) and \((-2)^{2}\) don't always mean the same thing.

f(2) = f(-2) and it is NOT one-to-one. Simple as that.

Answer 10

ok, could you give an example of something that would be a one-to-one?

Answer 11

y = x-4

Answer 12

y = x^3 - 5

Answer 13

y = e^x

Answer 14

oh ok, I think I see. in that first ex. if you had 2 for x then you would have y=2?

Answer 15

y = 2 is NOT one-to-one. Any value of x produces y = 2

If f(x) = 2, we have f(-3) = f(0) = f(5) = f(37) = 2

Answer 16

hmm, how can any value of x be y=2? if x = 3 then it would be y = (3)-4 which is y = -1 right? am I making this more complicated than it should be?

Answer 17

It was probably wrong of me to reuse f(x). I was using f(x) = 2 without remembering that we already used f(x) = x^2 - 7. My bad.

Answer 18

If g(x) = x - 7 the g(x) is one-to-one.

Answer 19

oh, because g(2) would be g(-2)=-5? which makes it a function right?

Answer 20

I did some research and remembered how to check it on the graph and it started making more sence. thanks for your patience @tkhunny :-)

Answer 21

Pass the Vertical Line test? Function.
ALSO pass the Horizontal Line test? one-to-one Function.