1. Problem ক: Given $g(t) = \frac{t^4 + t^2 + 1}{t^2}$, find $g(-3^{-1})$. 2. To solve, note that $-3^{-1} = -\frac{1}{3}$. 3. Substitute $t = -\frac{1}{3}$: $$g\left(-\frac{1}{3}\right) = \frac{\left(-\frac{1}{3}\right)^4 + \left(-\frac{1}{3}\right)^2 + 1}{\left(-\frac{1}{3}\right)^2}.$$ 4. Calculate each term: $$\left(-\frac{1}{3}\right)^4 = \frac{1}{81}, \quad \left(-\frac{1}{3}\right)^2 = \frac{1}{9}.$$ 5. So numerator: $$\frac{1}{81} + \frac{1}{9} + 1 = \frac{1}{81} + \frac{9}{81} + \frac{81}{81} = \frac{91}{81}.$$ 6. Denominator: $$\frac{1}{9}.$$ 7. Thus: $$g\left(-\frac{1}{3}\right) = \frac{\frac{91}{81}}{\frac{1}{9}} = \frac{91}{81} \times 9 = \frac{91}{9}.$$ 8. Problem খ: Show that $A \cup B = (A - B) \cup (B - A) \cup (A \cap B)$. 9. By definition, - $A \cup B$ contains all elements in $A$ or $B$. - $A - B$ elements in $A$ not in $B$. - $B - A$ elements in $B$ not in $A$. - $A \cap B$ elements in both. 10. Since union of these disjoint subsets covers all elements of $A$ and $B$, the equality holds. 11. Problem গ: Given $f(x) = \frac{4x+1}{4x-1}$ and equation $\frac{f(x+2)-1}{f(x-2)-1}+1 = 0$, find $x$. 12. Compute $f(x+2) - 1$: $$f(x+2) - 1 = \frac{4(x+2)+1}{4(x+2)-1} - 1 = \frac{4x + 8 + 1}{4x + 8 - 1} - 1 = \frac{4x + 9}{4x + 7} - 1 = \frac{4x + 9 - (4x + 7)}{4x + 7} = \frac{2}{4x + 7}.$$ 13. Compute $f(x-2) - 1$: $$f(x-2) - 1 = \frac{4(x-2)+1}{4(x-2)-1} - 1 = \frac{4x -8 +1}{4x -8 -1} - 1 = \frac{4x - 7}{4x - 9} - 1 = \frac{4x -7 - (4x -9)}{4x -9} = \frac{2}{4x -9}.$$ 14. Substitute into given equation: $$\frac{\frac{2}{4x + 7}}{\frac{2}{4x - 9}} + 1 = 0 \implies \frac{2}{4x + 7} \times \frac{4x - 9}{2} + 1 = 0.$$ 15. Simplify: $$\frac{4x - 9}{4x + 7} + 1 = 0 \implies \frac{4x - 9 + 4x + 7}{4x + 7} = 0 \implies \frac{8x - 2}{4x + 7} = 0.$$ 16. For fraction to be zero, numerator must be zero: $$8x - 2 = 0 \implies x = \frac{1}{4}.$$ Final answers: গ: $\boxed{\frac{1}{4}}$ ক: $\boxed{\frac{91}{9}}$ Slug: "set identity and functions" Subject: "set theory and algebra" Desmos: {"latex":"y=(4x+1)/(4x-1)","features":{"intercepts":true,"extrema":true}} q_count: 3