গণিত প্রশ্ন

উত্তর ও ব্যাখ্যা

1. ক)

   $f(x) = \dfrac{3x - 4}{5x - 8}$
   $f(-3) = \dfrac{3(-3) - 4}{5(-3) - 8} = \dfrac{-13}{-23} = \mathbf{\dfrac{13}{23}}$

2. খ)

   $f\left(\dfrac{1}{x^2}\right) = \dfrac{\frac{3}{x^2} - 4}{\frac{5}{x^2} - 8} = \dfrac{3 - 4x^2}{5 - 8x^2}$

   $\dfrac{f\left(\frac{1}{x^2}\right) - 1}{f\left(\frac{1}{x^2}\right) + 1} = \dfrac{(3 - 4x^2) - (5 - 8x^2)}{(3 - 4x^2) + (5 - 8x^2)} = \dfrac{4x^2 - 2}{8 - 12x^2} = \dfrac{2(2x^2 - 1)}{4(2 - 3x^2)} = \mathbf{\dfrac{2x^2 - 1}{2(2 - 3x^2)}}$

3. গ)

   $A = \{-2, -1, 0, 1, 2, 3\}$
   $S = \{(x, y) : x \in A,\ y \in A,\ y - x = 1\}$, অর্থাৎ $y = x + 1$

   $x = -2 \Rightarrow y = -1$; $x = -1 \Rightarrow y = 0$; $x = 0 \Rightarrow y = 1$; $x = 1 \Rightarrow y = 2$; $x = 2 \Rightarrow y = 3$; ($x = 3$ হলে $y = 4 \notin A$)

   সুতরাং $S = \{(-2, -1), (-1, 0), (0, 1), (1, 2), (2, 3)\}$
   রেঞ্জ $= \mathbf{\{-1, 0, 1, 2, 3\}}$