For sets $A$ and $B$, let $f : A \rightarrow B$ and $g : B \rightarrow A$ be functions such that $f ( g ( x ) ) = x$ for each $x$. For each statement below, write whether it is TRUE or FALSE. a) The function $f$ must be one-to-one.
Answer: $\_\_\_\_$ b) The function $f$ must be onto.
Answer: $\_\_\_\_$ c) The function $g$ must be one-to-one.
Answer: $\_\_\_\_$ d) The function $g$ must be onto.
Answer: $\_\_\_\_$

Given a real number $x$, define $g ( x ) = x ^ { 2 } e ^ { x }$ if $x \geq 0$ and $g ( x ) = x e ^ { - x }$ if $x < 0$.
(A) The function $g$ is continuous everywhere.
(B) The function $g$ is differentiable everywhere.
(C) The function $g$ is one-to-one.
(D) The range of $g$ is the set of all real numbers.

We are given a continuous function $\xi : \mathbb{R} \rightarrow \mathbb{R}$ satisfying condition (V.1) (with $d \geqslant 2$), where $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) = \left(\xi(A_{i,j})\right)_{1\leqslant i,j\leqslant d} \text{ invertible} \tag{V.1}$$
Deduce that the function $\xi$ is injective, then that it is strictly monotone on $\mathbb{R}$.

Let $f : \mathbb{R} \rightarrow \mathbb{R}^{2}$ be a function given by $f(x) = (x^{m}, x^{n})$, where $x \in \mathbb{R}$ and $m, n$ are fixed positive integers. Suppose that $f$ is one-one. Then
(a) Both $n$ and $m$ must be odd
(b) At least one of $m$ and $n$ must be odd
(c) Exactly one of $m$ and $n$ must be odd
(d) Neither $m$ nor $n$ can be odd.

Let $f(x) = \dfrac{x^2}{x-1}$. Which of the following is true?
(A) $f$ is neither one-one nor onto
(B) $f$ is one-one and onto
(C) $f$ is one-one but not onto
(D) $f$ is onto but not one-one

Let $\mathbb{R}$ be the set of all real numbers. The function $f : \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = x^3 - 3x^2 + 6x - 5$ is
(A) one-to-one, but not onto
(B) one-to-one and onto
(C) onto, but not one-to-one
(D) neither one-to-one nor onto.

Let $\mathbb{R}$ be the set of all real numbers. The function $f : \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = x^3 - 3x^2 + 6x - 5$ is
(A) one-to-one, but not onto
(B) one-to-one and onto
(C) onto, but not one-to-one
(D) neither one-to-one nor onto

Let $\mathbb { R }$ be the set of all real numbers. The function $f : \mathbb { R } \rightarrow \mathbb { R }$ defined by $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 6 x - 5$ is
(A) one-to-one, but not onto
(B) one-to-one and onto
(C) onto, but not one-to-one
(D) neither one-to-one nor onto

Consider the function $f : \mathbb { C } \rightarrow \mathbb { C }$ defined by $$f ( a + i b ) = e ^ { a } ( \cos b + i \sin b ) , a , b \in \mathbb { R }$$ where $i$ is a square root of $-1$. Then
(A) $f$ is one-to-one and onto.
(B) $f$ is one-to-one but not onto.
(C) $f$ is onto but not one-to-one.
(D) $f$ is neither one-to-one nor onto.

Let $\mathbb { R }$ be the set of all real numbers. The function $f : \mathbb { R } \rightarrow \mathbb { R }$ defined by $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 6 x - 5$ is
(a) one-to-one, but not onto.
(B) one-to-one and onto.
(C) onto, but not one-to-one.
(D) neither one-to-one nor onto.

If the function $f : \mathbb { R } \rightarrow \mathbb { R }$ is defined by $f ( x ) = | x | ( x - \sin x )$, then which of the following statements is TRUE?
(A) $f$ is one-one, but NOT onto
(B) $f$ is onto, but NOT one-one
(C) $f$ is BOTH one-one and onto
(D) $f$ is NEITHER one-one NOR onto

Let $\mathbb { N }$ denote the set of all natural numbers, and $\mathbb { Z }$ denote the set of all integers. Consider the functions $f : \mathbb { N } \rightarrow \mathbb { Z }$ and $g : \mathbb { Z } \rightarrow \mathbb { N }$ defined by
$$f ( n ) = \begin{cases} ( n + 1 ) / 2 & \text { if } n \text { is odd } \\ ( 4 - n ) / 2 & \text { if } n \text { is even } \end{cases}$$
and
$$g ( n ) = \begin{cases} 3 + 2 n & \text { if } n \geq 0 \\ - 2 n & \text { if } n < 0 \end{cases}$$
Define $( g \circ f ) ( n ) = g ( f ( n ) )$ for all $n \in \mathbb { N }$, and $( f \circ g ) ( n ) = f ( g ( n ) )$ for all $n \in \mathbb { Z }$.
Then which of the following statements is (are) TRUE?

(A)	$g \circ f$ is NOT one-one and $g \circ f$ is NOT onto
(B)	$f \circ g$ is NOT one-one but $f \circ g$ is onto
(C)	$g$ is one-one and $g$ is onto
(D)	$f$ is NOT one-one but $f$ is onto

Let $A$ and $B$ be non empty sets in $\mathbb{R}$ and $f : A \rightarrow B$ is a bijective function. Statement 1: $f$ is an onto function. Statement 2: There exists a function $g : B \rightarrow A$ such that $f \circ g = I _ { B }$.
(1) Statement 1 is true, Statement 2 is false.
(2) Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation for Statement 1.
(3) Statement 1 is false, Statement 2 is true.
(4) Statement 1 is true, Statement 2 is true, Statement 2 is not the correct explanation for Statement 1.

Let $f : R \rightarrow R$ be defined by $f ( x ) = \frac { | x | - 1 } { | x | + 1 }$, then $f$ is
(1) one-one but not onto
(2) neither one-one nor onto
(3) both one-one and onto
(4) onto but not one-one

The function $f : \mathbb { R } \to \left[ - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right]$ defined as $f ( x ) = \frac { x } { 1 + x ^ { 2 } }$, is:
(1) Surjective but not injective
(2) Neither injective nor surjective
(3) Invertible
(4) Injective but not surjective

The function $f : \mathbb{R} \rightarrow \left(-\dfrac{1}{2}, \dfrac{1}{2}\right)$ defined as $f(x) = \dfrac{x}{1 + x^2}$, is:
(1) Invertible
(2) Injective but not surjective
(3) Surjective but not injective
(4) Neither injective nor surjective

Let $A = \{x \in R : x$ is not a positive integer$\}$. Define a function $f: A \rightarrow R$ as $f(x) = \frac{2x}{x-1}$, then $f$ is:
(1) Injective but not surjective
(2) Not injective
(3) Surjective but not injective
(4) Neither injective nor surjective

If the function $f : R - \{ 1 , - 1 \} \rightarrow A$ defined by $f ( x ) = \frac { x ^ { 2 } } { 1 - x ^ { 2 } }$, is surjective, then $A$ is equal to
(1) $[ 0 , \infty )$
(2) $R - \{ - 1 \}$
(3) $R - [ - 1,0 )$
(4) $R - ( - 1,0 )$

Let $f : R \rightarrow R$ be defined as $f(x) = 2x - 1$ and $g : R - \{1\} \rightarrow R$ be defined as $g(x) = \frac { x - \frac { 1 } { 2 } } { x - 1 }$. Then the composition function $f(g(x))$ is:
(1) neither one-one nor onto
(2) onto but not one-one
(3) both one-one and onto
(4) one-one but not onto

Consider function $f : A \rightarrow B$ and $g : B \rightarrow C ( A , B , C \subseteq R )$ such that $( g o f ) ^ { - 1 }$ exists, then:
(1) $f$ and $g$ both are one-one
(2) $f$ and $g$ both are onto
(3) $f$ is one-one and $g$ is onto
(4) $f$ is onto and $g$ is one-one

If $f: R \rightarrow R$ is a function defined by $f(x) = e^{|x|} - e^{-x}$ / $e^{x} + e^{-x}$, then $f$ is:
(1) bijective
(2) $f$ is monotonically increasing on $(0, \infty)$
(3) $f$ is monotonically decreasing on $(0, \infty)$
(4) not differentiable at $x = 0$

Let $f , g : N \rightarrow N$ such that $f ( n + 1 ) = f ( n ) + f ( 1 ) \forall n \in N$ and $g$ be any arbitrary function. Which of the following statements is NOT true?
(1) If $f$ is onto, then $f ( n ) = n \forall n \in N$
(2) If $g$ is onto, then $f o g$ is one-one
(3) $f$ is one-one
(4) If $f \circ g$ is one-one, then $g$ is one-one

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be defined as $f ( x ) = x - 1$ and $g : R \rightarrow \{ 1 , - 1 \} \rightarrow \mathbb { R }$ be defined as $g ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } - 1 }$. Then the function $f o g$ is:
(1) One-one but not onto
(2) onto but not one-one
(3) Both one-one and onto
(4) Neither one-one nor onto

Let a function $f : \mathbb{N} \rightarrow \mathbb{N}$ be defined by $$f(n) = \begin{cases} 2n, & n = 2,4,6,8,\ldots \\ n-1, & n = 3,7,11,15,\ldots \\ \frac{n+1}{2}, & n = 1,5,9,13,\ldots \end{cases}$$ then, $f$ is
(1) One-one and onto
(2) One-one but not onto
(3) Onto but not one-one
(4) Neither one-one nor onto

Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined by $f(x) = \frac{x^2 + 2}{x^2 + 1}$. Then which of the following is NOT true?
(1) $f(x)$ has a minimum at $x = 0$
(2) $f(x)$ is an even function
(3) $f(x)$ is strictly increasing for $x > 0$
(4) $f(x)$ is onto