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Identity Function

Which of the ...

Question

Which of the following functions are identity functions?

A

f : R \to R, f (x) = x

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B

g : N \to Z, g (p) = 3

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C

h : z \to z, h (y) = y

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D

g : N \to N, g (z) = z

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Solution

The correct options are
A $f : R \to R, f (x) = x$
C $h : z \to z, h (y) = y$
D $g : N \to N, g (z) = z$
An identity function is a function that always returns the same value that was used as its argument.

Hence $f : R \to R, f (x) = x$ is an identity function

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Similar questions

f : Z \to Z

g : Z \to Z

be functions defined by

f = {(n, n^{2}) : n \in Z}

g : {(n, | n |^{2}) : n \in Z}

f = g

.

Q. Consider f : N → N, g : N → N and h : N → R defined as f(x) = 2x, g(y) = 3y + 4 and h(z) = sin z for all x, y, z ∈ N. Show that ho (gof) = (hog) of.

f : X \to Y

g : Y \to Z

h : Z \to S

are functions, then

h o (g o f) = (h o g) o f

Q. Classify the following functions as injection, surjection or bijection :
(i) f : N → N given by f(x) = x²
(ii) f : Z → Z given by f(x) = x²
(iii) f : N → N given by f(x) = x³
(iv) f : Z → Z given by f(x) = x³
(v) f : R → R, defined by f(x) = |x|
(vi) f : Z → Z, defined by f(x) = x² + x
(vii) f : Z → Z, defined by f(x) = x − 5
(viii) f : R → R, defined by f(x) = sinx
(ix) f : R → R, defined by f(x) = x³ + 1
(x) f : R → R, defined by f(x) = x³ − x
(xi) f : R → R, defined by f(x) = sin²x + cos²x
(xii) f : Q − {3} → Q, defined by

f (x) = \frac{2 x + 3}{x - 3}

(xiii) f : Q → Q, defined by f(x) = x³ + 1
(xiv) f : R → R, defined by f(x) = 5x³ + 4
(xv) f : R → R, defined by f(x) = 3 − 4x
(xvi) f : R → R, defined by f(x) = 1 + x²
(xvii) f : R → R, defined by f(x) =

\frac{x}{x^{2} + 1}

[NCERT EXEMPLAR]

Q. 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

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