MCQ
The function $f(x)=\frac{x^2+2 x-15}{x^2-4 x+9}, x \in R$ is
- Aboth one $-$ one and onto.
- Bonto but not one $-$ one.
- ✓neither one $-$ one nor onto.
- Done $-$ one but not onto.
✓
Answer
Correct option: C.
neither one $-$ one nor onto.
$f(x)=\frac{(x+5)(x-3)}{x^2-4 x+9}$
Let $g(x)=x^2-4 x+9$
$ D < 0 $
$ g(x) > 0$ for $x \in R$
$\therefore\left[\begin{array}{l}\mathrm{f}(-5)=0 \\ \mathrm{f}(3)=0\end{array}\right.$
So, $\mathrm{f}(\mathrm{x})$ is many $-$ one.
again,
$ y x^2-4 x y+9 y=x^2+2 x-15 $
$ x^2(y-1)-2 x(2 y+1)+(9 y+15)=0 $
$ \text { for } \forall x \in R$
$ \Rightarrow D \geq 0 $
$ D=4(2 y+1)^2-4(y-1)(9 y+15) \geq 0 $
$ 5 y^2+2 y+16 \leq 0 $
$ (5 y-8)(y+2) \leq 0$
Image
$\mathrm{y} \in\left[-2, \frac{8}{5}\right]$ range
Note : If function is defined from $f: R \rightarrow R$ then only correct answer is option ($3$)
$\Rightarrow$ Bonus
Let $g(x)=x^2-4 x+9$
$ D < 0 $
$ g(x) > 0$ for $x \in R$
$\therefore\left[\begin{array}{l}\mathrm{f}(-5)=0 \\ \mathrm{f}(3)=0\end{array}\right.$
So, $\mathrm{f}(\mathrm{x})$ is many $-$ one.
again,
$ y x^2-4 x y+9 y=x^2+2 x-15 $
$ x^2(y-1)-2 x(2 y+1)+(9 y+15)=0 $
$ \text { for } \forall x \in R$
$ \Rightarrow D \geq 0 $
$ D=4(2 y+1)^2-4(y-1)(9 y+15) \geq 0 $
$ 5 y^2+2 y+16 \leq 0 $
$ (5 y-8)(y+2) \leq 0$
Image
$\mathrm{y} \in\left[-2, \frac{8}{5}\right]$ range
Note : If function is defined from $f: R \rightarrow R$ then only correct answer is option ($3$)
$\Rightarrow$ Bonus
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Let P the point of intersection of the lines
Then, the shortest distance of P from the line $4x = 2y = z$ is
→Then, the shortest distance of P from the line $4x = 2y = z$ is
Consider the integral
→$I=\int_{0}^{10} \frac{[x] e^{[x]}}{e^{x-1}} d x,$
where $[ x ]$ denotes the greatest integer less than or equal to $x$. Then the value of $I$ is equal to:
The order and degree of the differential equation $\frac{{{d^2}y}}{{d{x^2}}} = \sqrt {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} $ is
→If the graph of $y = ax^3 + bx^2 + cx + d$ is symmetric about the line $x = k$ then
→If $n$ is the smallest natural number such that $n+2 n+3 n+\ldots+99 n$ is a perfect square, then the number of digits of $n^2$ is
→$\int_{}^{} {\frac{{2x{{\tan }^{ - 1}}{x^2}}}{{1 + {x^4}}}} \;dx = $
→In the expansion of ${(1 + 3x + 2{x^2})^6}$ the coefficient of ${x^{11}}$ is
→The number of real values of $\lambda $ for which the system of linear equations $2x + 4y - \lambda z = 0$ ;$4x + \lambda y + 2z = 0$ ; $\lambda x + 2y+ 2z = 0$ has infinitely many solutions, is
→Let $U = \{ 1,\,2,\,3,\,4,\,5,\,6,\,7,\,8,\,9,\,10\} $, $A = \{ 1,\,2,\,5\} ,\,B = \{ 6,\,7\} $, then $A \cap B'$ is
→Let $z _{1}$ and $z _{2}$ be two complex numbers such that $\overline{ z }_{1}=i \overline{ z }_{2}$ and $\arg \left(\frac{ z _{1}}{\overline{ z }_{2}}\right)=\pi$. Then
→