Question:
The function \(f : \mathbf{N} \to \mathbf{N}\) defined by \(f(x) = x - 5\left[\frac{x}{5}\right]\), where \(\mathbf{N}\) is the set of natural numbers and \([x]\) denotes the greatest integer less than or equal to \(x\), is :
The function \(f : \mathbf{N} \to \mathbf{N}\) defined by \(f(x) = x - 5\left[\frac{x}{5}\right]\), where \(\mathbf{N}\) is the set of natural numbers and \([x]\) denotes the greatest integer less than or equal to \(x\), is :
Show Hint
f(x) is the remainder when x is divided by 5, so it repeats values and never reaches large numbers.
Updated On: Nov 11, 2026
- one-one and onto.
- one-one but not onto.
- onto but not one-one.
- neither one-one nor onto.
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
\(f(x) = x - 5[x/5]\) is the remainder of \(x\) on division by 5, so its values lie in \(\{0,1,2,3,4\}\).
\(f(1) = 1 - 0 = 1\) and \(f(6) = 6 - 5 = 1\). Two different inputs give the same output, so \(f\) is not one-one.
The range is contained in \(\{0,1,2,3,4\}\), which does not contain every natural number (for example 6 has no pre-image), so \(f\) is not onto.
Answer: (4) neither one-one nor onto.
\(f(1) = 1 - 0 = 1\) and \(f(6) = 6 - 5 = 1\). Two different inputs give the same output, so \(f\) is not one-one.
The range is contained in \(\{0,1,2,3,4\}\), which does not contain every natural number (for example 6 has no pre-image), so \(f\) is not onto.
Answer: (4) neither one-one nor onto.
Was this answer helpful?
0
0
Top JEE Main Mathematics Questions
- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
- If , then the general value of is:
- The general solution of
- If the constant term in the binomial expansion of $\left(\frac{x^{\frac{3}{2}}}{2}-\frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3 l}$ is $2^\alpha \beta$, where $\beta<0$ is an odd number, then $|\alpha l-\beta|$ is equal to_____
View More Questions
Top JEE Main types of functions Questions
- The number of bijective functions $f :\{1,3,5$, $7, \ldots \ldots 99\} \rightarrow\{2,4,6,8, \ldots \ldots , 100\}$, such that $f (3) \geq f (9) \geq f (15) \geq f (21) \geq \ldots f (99), $ is _____
- If the functions $f(x)=\frac{x^3}{3}+2 b x+\frac{a x^2}{2}$ and $g(x)=\frac{x^3}{3}+a x+b x^2, a \neq 2 b$ have a common extreme point, then $a+2 b+7$ is equal to:
- If the function ƒ defined on \( \left( { - {1 \over 3},{1 \over 3}} \right) \) by
f(x) = \( \left\{ {\matrix{ {{1 \over x}{{\log }_e}\left( {{{1 + 3x} \over {1 - 2x}}} \right),} & {when\,x \ne 0} \cr {k,} & {when\,x = 0} \cr } } \right. \)
is continuous, then k is equal to_______. - Let \(f\) be a composite function of \(x\) defined by \(f(u)=\frac{1}{u^2+u-2},\ u(x)=\frac{1}{x-1}\). Then the number of points \(x\) where \(f\) is discontinuous is :
- Let \(A=\{1,2,3,4\}\) and \(R:A\to A\) be the relation defined by \(R=\{(1,1),(2,3),(3,4),(4,2)\}\). The correct statement is :
View More Questions
Top JEE Main Questions
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
A black body is at a temperature of 2880 K. The energy of radiation emitted by this body with wavelength between 499 nm and 500 nm is U1, between 999 nm and 1000 nm is U2 and between 1499 nm and 1500 nm is U3. The Wien's constant, b = 2.88×106 nm-K. Then,
View More Questions