$\begin{aligned}
& \operatorname{fog}(x)=f(g(x)) \
& =f\left(\frac{2-3 x}{1-x}\right)=\frac{2\left(\frac{2-3 x}{1-x}\right)+3}{5\left(\frac{2-3 x}{1-x}\right)+2} \
& =\frac{4-6 x+3-3 x}{10-15 x+2-2 x}=\left(\frac{7-9 x}{12-17 x}\right) \
& \therefore\left[\begin{array}{c}
12-7 x \neq 0 \
x \neq \frac{12}{17}
\end{array}\right. \
& {\left[\begin{array}{l}
\operatorname{fog}(2)=\frac{7-9(2)}{12-17(2)}=\frac{-11}{-22}=\frac{1}{2} \
\operatorname{fog}(4)=\frac{7-9(4)}{12-17(4)}=\frac{-29}{-56}=\frac{29}{56}
\end{array}\right.} \
& \text { Range of fog : }[\alpha, \beta]=\left[\frac{1}{2}, \frac{29}{56}\right] \
& \therefore(\beta-\alpha)=\frac{29}{56}-\frac{1}{2}=\frac{29-28}{56}=\frac{1}{56} \
& \frac{1}{(\beta-\alpha)}=56
\end{aligned}$
Let f, ~g:(1, ∞) → mathbbR be defined as f( x)= 2 x+3 5 x+2 and g(x)= 2-3 x 1-x. If the range of the function f ° g:[2,4] → mathbbR is [α, β], then 1…
JEE Main 2025 — Mathematics Algebra
2025mcqmedium
Let f, g:(1,∞)→R be defined as f(x)=5x+22x+3 and g(x)=1−x2−3x. If the range of the function f∘g:[2,4]→R is [α,β], then β−α1 is equal to
Official previous-year question
Held on 4 Apr 2025 · Verified 6 Jul 2026.
Options
- A
68
- B
29
- C
2
- D
56
Did you get this right?
Sign in to track your attempts and accuracy.
Your note
Sign in to keep a private note on this question. Nothing you write is ever public.
JEE Main Algebra in other years
More JEE Main Algebra questions
The number of elements in the set $S = \left\{(r, k) : k \in \mathbb{Z} \text{ and } {}^{36}C_{r+1} = \dfrac{6\left({}^{35}C_r\right)}{(k^2 - 3)}\right\}$, is :
2026medium
Suppose $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in A.P. and $\mathrm{a}^{2}, 2 \mathrm{~b}^{2}, \mathrm{c}^{2}$ are in G.P. If $\mathrm{a}<\mathrm{b}<\mathrm{c}$ and $\mathrm{a}+\mathrm{b}+\mathrm{c}=1$, then $9\left(\mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}\right)$ is equal to $\_\_\_\_$ .
2026medium
Let $S=\left\{z \in \mathbb{C}: 4 z^{2}+\bar{z}=0\right\}$. Then $\sum_{z \in S}|z|^{2}$ is equal to:
2026medium
Let $S=\{z: 3 \leqslant|2 z-3(1+i)| \leqslant 7\}$ be a set of complex numbers. Then $\min _{z \in S}\left|\left(z+\frac{1}{2}(5+3 i)\right)\right|$ is equal to :
2026medium
Let $A = \begin{bmatrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 9 & 3 & 1 \end{bmatrix}$ and $B = [b_{ij}]$, $1 \leq i, j \leq 3$. If $B = A^{99} - I$, then the value of $\dfrac{b_{31} - b_{21}}{b_{32}}$ is :
2026medium
Other Mathematics topics for JEE Main
Keep practicing Algebra
Work through every JEE Main Algebra PYQ, year by year.