1. Determine whether each of these functions is a bijection...

1. Determine whether each of these functions is a bijection from R to R. [6 Points] a. f(x) = x^2 + 1 b. f(x) = (x^2 + 1)/(x^2 + 2) c. f(x) = x^3 d. f(x) = 3x + 4 e. f(x) = x^3 + 1 f. f(x) = |x|

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Transcript

00:01 Okay, so we have these functions and we want to determine whether they're bijections from r to r.

00:06 So the first function, a, is f of x equals x squared plus 1.

00:11 And the answer here is no, because it's not injective or subjective.

00:15 So let's just show that it's not injective.

00:17 F of 1 equals 1 square plus 1, which is 2.

00:21 F of minus 1 equals minus 1 squared plus 1, which is 1, which is 2.

00:28 So we have that f of 1 equals f of minus 1, and therefore it's not injective because two points, two different points mapped to the same point.

00:38 For part b, f of x equals x square plus 1 over x square plus 2, and it's not bijective for a similar reason.

00:47 F of 1, for example, is 1 square plus 1 on the top 2, 1 square plus 2 is 3, f of minus 1 is minus 1 squared, which is 1 plus 1, which is 2.

01:00 And then on the bottom, we have minus 1 squared, which is 1 plus 2, which is 3.

01:05 So again, we have two different points.

01:07 Mapping to the same point, so not injective.

01:11 For part c, f of x equals x cubed.

01:14 So this is injective.

01:17 So this is injective and subjective, sorry.

01:20 So it's bijective, and it has inverse function.

01:23 F inverse.

01:24 We can just take the cubed root, and this is well defined for each number...

Question

1. Determine whether each of these functions is a bijection from R to R. [6 Points] a. f(x) = x^2 + 1 b. f(x) = (x^2 + 1)/(x^2 + 2) c. f(x) = x^3 d. f(x) = 3x + 4 e. f(x) = x^3 + 1 f. f(x) = |x|

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