Show that the function f : R_{*} → R_{*} defined by f(x) = 1/x is one-one and onto,where R_{*} is the set of all non-zero real numbers. Is the result true, if the domain R_{*} is replaced by N with co-domain being same as R_{* }?

Check the injectivity and surjectivity of the following functions:

(i) f : N → N given by f(x) = x^{2}

(ii) f : Z → Z given by f(x) = x^{2}

(iii) f : R → R given by f(x) = x^{2}

(iv) f : N → N given by f(x) = x^{3}

(v) f : Z → Z given by f(x) = x^{3 }

Prove that the Greatest Integer Function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

Show that the Modulus Function f : R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative.

Show that the Signum Function f : R → R, given by

is neither one-one nor onto.

Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. Show that f is one-one.

In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.

(i) f : R → R defined by f(x) = 3 – 4x

(ii) f : R → R defined by f(x) = 1 + x^{2 }

Let A and B be sets. Show that f : A × B → B × A such that f(a, b) = (b, a) is bijective function.

Let f : N → N be defined by State whether the function f is bijective. Justify your answer.

Let A = R – {3} and B = R – {1}. Consider the function f : A → B defined by

Let f : R → R be defined as f(x) = x^{4}. Choose the correct answer.

(A) f is one-one onto

(B) f is many-one onto

(C) f is one-one but not onto

(D) f is neither one-one nor onto.

Let f : R → R be defined as f(x) = 3x. Choose the correct answer.

Let f : {1, 3, 4} → {1, 2, 5} and g : {1, 2, 5} → {1, 3} be given by f = {(1, 2), (3, 5), (4, 1)} and g = {(1, 3), (2, 3), (5, 1)}. Write down gof.

Let f, g and h be functions from R to R. Show that

(f + g)oh = foh + goh

(f . g)oh = (foh) . (goh)

Find gof and fog, if

(i) f(x) = | x | and g(x) = | 5x – 2 | (ii) f(x) = 8x3 and g(x) = x^{1/3} .

If f(x) = _{}, show that fof(x) = x, for all x ≠ 2/3. What is the inverse of f ?

State with reason whether following functions have inverse

(i) f : {1, 2, 3, 4} → {10} with

f = {(1, 10), (2, 10), (3, 10), (4, 10)}

(ii) g : {5, 6, 7, 8} → {1, 2, 3, 4} with

g = {(5, 4), (6, 3), (7, 4), (8, 2)}

(iii) h : {2, 3, 4, 5} → {7, 9, 11, 13} with

h = {(2, 7), (3, 9), (4, 11), (5, 13)}

Show that f : [–1, 1] → R, given by is one-one. Find the inverse of the function f : [–1, 1] → Range f.

(Hint: For y ∈ Range f, y =, for some x in [ - 1, 1], i.e.,)

Consider f : R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.

Consider f : R_{+} → [4, ∞) given by f(x) = x^{2} + 4. Show that f is invertible with the inverse f^{–1} of f given by _{}, where R_{+} is the set of all non-negative real numbers.

Consider f : R_{+} → [– 5, ∞) given by f(x) = 9x^{2} + 6x – 5. Show that f is invertible with .

Let f : X → Y be an invertible function. Show that f has unique inverse.

(Hint: suppose g1 and g2 are two inverses of f. Then for all y ∈ Y, fog1(y) = 1Y(y) = fog2(y). Use one-one ness of f).

Consider f : {1, 2, 3} → {a, b, c} given by f(1) = a, f(2) = b and f(3) = c. Find f^{ –1} and show that (f^{ –1})^{–1} = f.

Let f: X → Y be an invertible function. Show that the inverse of f ^{–1} is f, i.e., (f^{–1})^{–1} = f.

If f: R → R be given by f(x) = _{}, then fof(x) is (A) _{}

(B) x^{3}

(C) x

(D) (3 – x^{3}).

Letbe a function defined as. The inverse of f is map g: Range

(A)

(B)

(C)

(D)