Let $[\cdot]$ denote the greatest integer function. If $f(x) = [x]$ and $g(x) = 3[\frac{x}{3}]$, then the set of all real $x$ such that $f(x) = g(x)$ is
- A$R$
- B$\{x \in R : x = 3k, k \in Z\}$
- C$\{x \in R : 3k - 1 < x \leq 3k, k \in Z\}$
- D$\{x \in R : 3k \leq x < 3k + 1, k \in Z\}$
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Medium
View SolutionWhich of the following functions is surjective but not injective?
Let $R$ denote the set of all real numbers and $R^{+}$ denote the set of all positive real numbers. For the subsets $A$ and $B$ of $R$,define $f: A \rightarrow B$ by $f(x) = x^2$ for $x \in A$. Match the items in Column-$I$ with the items in Column-$II$.
Column-$I$ Column-$II$ $A$. $f$ is one-one and onto,if $1$. $A = R^{+}, B = R$ $B$. $f$ is one-one but not onto,if $2$. $A = B = R$ $C$. $f$ is onto but not one-one,if $3$. $A = R, B = R^{+}$ $D$. $f$ is neither one-one nor onto,if $4$. $A = B = R^{+}$
| Column-$I$ | Column-$II$ |
|---|---|
| $A$. $f$ is one-one and onto,if | $1$. $A = R^{+}, B = R$ |
| $B$. $f$ is one-one but not onto,if | $2$. $A = B = R$ |
| $C$. $f$ is onto but not one-one,if | $3$. $A = R, B = R^{+}$ |
| $D$. $f$ is neither one-one nor onto,if | $4$. $A = B = R^{+}$ |
DifficultTS EAMCET 2010
View SolutionLet $f: R \to R$ be defined as $f(x) = x^3$. Then $f$ is . . . . . . .
MediumGUJCET 2025
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