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Definition of Function

Find whether ...

Question

Find whether the following function are onto or not:
$f : Z \to Z$ defined by $f (x) = 6 x - 7$ for all $x ϵ Z$

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Solution

$f : Z \to Z$

$f (x) = 6 x - 7 \to$ subjective function

$f^{'} (x) = 6$ \rightarrow \therefore$ one-one function

$∴$ $f (x)$ is an onto function.

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Similar questions

Q. Find whether the following functions are one-one , onto or not :

f : R \to R

f (x) = x^{2} + 3

x ϵ R

f : Z \to Z

f (x) = x^{2}, x ϵ Z

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. Show that the function

f : Z \to Z

f (x) = x^{2} + x \forall x \in Z

, is a many-one function.

Q. State whether the function

f : R \to R

f (x) = 1 + x^{2}

is one-one onto or bijective.

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