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Standard XII

Mathematics

Definition of Function

The function ...

Question

The function $f : R \to R$ is defined by $f (x) = 3^{- x}$ . Then observe the following statements
I. $f$ is one-one function
II. $f$ is onto
III. $f$ is a decreasing function
Out of these, true statements are :

only I, II

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only II, III

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only I, III

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I, II, III

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Solution

The correct option is B only I, III
$f (x) = 3^{- x}$
Range of $f (x)$ is $(0, \infty)$
Here range of $f (x)$ $\neq$ codomain, hence, the function is not onto
Also $f (x)$ is monotonic decreasing function.
And we know every monotonic function is one - one function
so (I), (III) are true statements.

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Similar questions

Q. The function

f : R \to R

is defined by

f (x) = 3^{- x}

. Observe the following statements
I.

f

is one-one.
II.

f

is onto.
III.

f

is a decreasing function.
Out of these, true statements are

f : R \to R

is defined as

f (x) = 3^{x}

then which of the following is correct
I)

f

is one-one
II)

f

is onto
III)

f

is a increasing function

Q. A function is defined as

f (x) = a^{x} \forall ϵ R

and 0 < a < 1, then which of the following is true ?
(i) It is an increasing function
(ii) It is an decreasing function
(iii) All such functions must meet at (0, 1)

Q. I: If

f : A \to B

is a bijection only then does

f

have an inverse function
II: The inverse function

f : R^{+} \to R^{+}

defined by

f (x) = x^{2}

f^{- 1} (x) = \sqrt{x}

Q. Let

R

be the set of real numbers and

f : R \to R

be defined by

f (x) = \frac{{x}}{1 + [x]^{2}},

where

[x]

is the greatest integer less than or equal to

x,

and

{x} = x - [x] .

Which of the following statements are true?

I. The range of

f

is a closed interval.
II.

f

is continuous on

R .

III.

f

is one-one on

R

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The function f:R→R is defined by f(x)=3−x. Then observe the following statements I. f is one-one functionII. f is ontoIII. f is a decreasing functionOut of these, true statements are :

The correct option is B only I, IIIf(x)=3−xRange of f(x) is (0,∞)Here range of f(x) ≠ codomain, hence, the function is not ontoAlso f(x) is monotonic decreasing function.And we know every monotonic function is one - one functionso (I), (III) are true statements.

The function $f : R \to R$ is defined by $f (x) = 3^{- x}$ . Then observe the following statements
I. $f$ is one-one function
II. $f$ is onto
III. $f$ is a decreasing function
Out of these, true statements are :

The correct option is B only I, III
$f (x) = 3^{- x}$
Range of $f (x)$ is $(0, \infty)$
Here range of $f (x)$ $\neq$ codomain, hence, the function is not onto
Also $f (x)$ is monotonic decreasing function.
And we know every monotonic function is one - one function
so (I), (III) are true statements.