1. For each of the following, (i) (1 pt) Determine whether each of the following

Question

1. For each of the following, (i) (1 pt) Determine whether each of the following is a function. If not, justify your answer. If so, answer questions (ii-iv). You do not need to write any proofs here. (ii) (1 pt) Is the function 1-1? (iii) (1 pt) Is the function onto? (iv) (1 pt) ls the function invertible? If so, write the inverse function. a. f R-R such that f(x) 1/z b. f RtR+ such that f(x)-1/r c. f:R-R such that f(x)-V2x2 +1 d, f : R ? R such that f(x) = 3x-4 e, f : × Z ? Z such that f(m, n)-mn

Explanation / Answer

(a) f is not a function from R to R, since 0 has no image under f

(b) f is a function from R+ to R+, f is one-one, onto & invertible,

let, f(x)=f(y), then, 1/x = 1/y => x=y ,i.e. f is one-one,

again, for all x in the codomain there is 1/x in the domain such that, f(1/x) = x, so f is onto.

since f is both one one & onto, f is a bijection, hence invertible. and, f-1 (x) = 1/x as well.

(c) f is a function from R to R, f is not one-one, not onto & not invertible,

let, f(x)=f(y), then, (2x2 + 1)1/2 = (2y2 + 1)1/2 => x2 = y2 ,i.e. x = +- y , so f is not one-one,

again, -1 has no pre-image, so f is not onto.

since f is not both one one & onto, f is not a bijection, hence not invertible.

(d) f is a function from R to R, f is one-one, onto & invertible,

let, f(x)=f(y), then, 3x-4 = 3y-4 => x=y ,i.e. f is one-one,

again, for all x in the codomain there is (x+4)/3 in the domain such that, f((x+4)/3) = x, so f is onto.

since f is both one one & onto, f is a bijection, hence invertible. and, f-1 (x) = (x+4)/3

(e) f is a function from R to R, f is not one-one, but onto & not invertible,

let, f(3,4) = 12 = f(2,6) = f(12,1) ,i.e. f is not one-one,

again, for all m in the codomain there is (m,1) in the domain such that, f(m,1) = m, so f is onto.

since f is not one one, f is not a bijection, hence not invertible.

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