Let the function f : R - -b → R - 1 be defined by f(x)= x+a x+b ,… - Vidyadip

MCQ

Let the function f : R - {-b} → R - {1} be defined by $\text{f(x)}=\frac{\text{x}+\text{a}}{\text{x}+\text{b}},\ \text{a}\neq\text{b}.$ Then,

A
f is one-one but not onto.
B
f is onto but not one-one.
✓
f is both one-one and onto.
D
None of these.

✓

Answer

Correct option: C.

f is both one-one and onto.

Injectivity: Let x and y be two elements in the domain R - {-b}, such that

f(x) = f(y) ⇒ x + ax + b = y + ay + b

⇒ x + ay + b = x + by + a

⇒ xy + bx + ay + ab = xy + ax + by + ab

⇒ bx + ay = ax + by

⇒ a - bx = a - by

⇒ x = y

So, f is one-one.

Surjectivity: Let y be an element in the co-domain of f,

i.e., R - {1}, such that f(x) = y

⇒ x + ax + b = y

⇒ x + a ⇒ x = -a

So, f is onto.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from RELATIONS AND FUNCTIONS→RELATIONS AND FUNCTIONS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

The derivative of $\sec^{-1}\Big(\frac{1}{2\text{x}^2+1}\Big)$ w.r.t. $\sqrt{1+3\text{x}}$ at $\text{x}=\frac{-1}{3}$

Let $f(\mathrm{x})=\mathrm{x}^5+2 \mathrm{e}^{\mathrm{x} / 4}$ for all $\mathrm{x} \in \mathrm{R}$. Consider a function $g(x)$ such that $(gof) (x)=x$ for all $x \in R$. Then the value of $8 g^{\prime}(2)$ is :

If area bounded by the curves $x = at^2$ and $y = ax^2$ is $1,$ then a $.......$

Let $P, Q, R$ and $S$ be the points on the plane with position vectors $-2 \hat{i}-\hat{j}, 4 \hat{i}, 3 \hat{i}+3 \hat{j}$ and $-3 \hat{i}+2 \hat{j}$ respectively. The quadrilateral $\mathrm{PQRS}$ must be a

In a regular hexagon $\text{ABCDEF}, \overrightarrow{\text{AB}}=\vec{\text{a}},\ \overrightarrow{\text{BC}}=\vec{\text{b}}$ and $\overrightarrow{\text{CD}}=\vec{\text{c}}$. Then, $\overrightarrow{\text{AE}}=$

The radius of a circular plate is increasing at the rate of 0.01cm/sec. The rate of increase of its area when the radius is 12cm, is:

If the system of linear equations, $x+y+z =6$ ; $x+2 y+3 z =10$ ; $3 x+2 y+\lambda z =\mu$ has more two solutions, then $\mu-\lambda^{2}$ is equal to

Find the inverse of each of the matrices (if it exists). $\left[\begin{array}{ccc}1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4\end{array}\right]$

The area bounded by $y = 2 - x^2$ and $x + y = 0$ is :

If $A=\left[\begin{array}{rr}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right]$, then for what value of $\alpha, A$ is an identity matrix?