Functions
Range of rational function
nta_pyq_2025_apr
Grade 12
Question:
If the range of the function f$(x) = 2$$5-x$, x$\ne$1, 2, is (-$\infty$,$\alpha$]$\cup$[$\beta$,$\infty$), then$\alpha$+$\beta$is equal to : 2 2$x -3x+2$
Step-by-Step Solution
Key Concept: Rewrite the function to isolate the variable and determine all possible output values.
$y = 5 - x$2$x - 3x + 2$(4) 2$yx - 3xy + 2y + x - 5 = 0$2$yz + (-3y + 1)x + (2y - 5) = 0$Case I : If$y = 0$(Accepted) $\Rightarrow$$x = 5$Case II : If y$\ne$0 D$\ge$0 2$(-3y + 1) - 4(y)(2y - 5)$$\ge$0 2 2$9y + 1 - 6y - 8y + 20y$$\ge$0 2$y + 14y + 1$$\ge$0 2$(y + 7) - 48$$\ge$0$|y + 7|$$\ge$4$\sqrt{3}$$\Rightarrow$$y + 7$$\ge$4$\sqrt{3}$or$y + 7$$\le$-4$\sqrt{3}$$\Rightarrow$ y$\ge$4$\sqrt{3}$- 7 or y$\le$-4$\sqrt{3}$- 7 From Case I and Case II y$\ in $(-$\infty$, -4$\sqrt{3}$- 7]$\cup$[4$\sqrt{3}$- 7,$\infty$) So$\alpha$= -4$\sqrt{3}$- 7$\beta$= 4$\sqrt{3}$- 7 2 2 2 2 $\Rightarrow$$a + b = ($-4$\sqrt{3}$-$7) + ($4$\sqrt{3}$-$7) = 2(48 + 49) = 194$
Correct Answer: 4