Let the domain of the function be [$\alpha$,$\beta$] and the domain of$g(x) = log$$(2 - 6$log be ($\gamma$,$\delta$)$. -1$$4x+5$f$(x) = cos$( )$(2x + 5))$$3x-7$2 27 Then |7($\alpha$+$\beta$) + 4($\gamma$+$\delta$)| is equal to ________
Step-by-Step Solution
Key Concept: Impose every domain restriction from logarithms, roots, denominators, and inverse functions before intersecting intervals.
f$(x) = cos -1$($4x + 5$) (96)$3x - 7$$4x + 5$$\Rightarrow$ -1$\le$( )$\le$1$3x - 7$$4x + 5$( )$\ge$-1$3x - 7$$4x + 5 + 3x - 7$$\ge$0$3x - 7$$7x - 2$$\Rightarrow$$\ge$0$3x - 7$2 7 x$\ in $(-$\infty$, ]$\cup$( ,$\infty$) 7 3$4x + 5$$x + 12$&$\le$1 $\Rightarrow$$\le$0$3x - 7$$3x - 7$∴ Domain of f (x) is 2 2 [-12, ]$\alpha$= -12,$\beta$= 7 7$g(x) = log$$(2 - 6$log$(2x + 5))$2 27 Domain$2 - 6$log$(2x + 5) > 0$27 $\Rightarrow$ 6 log$(2x + 5) < 2$27 1 $\Rightarrow$ log$(2x + 5) < 27$3 $\Rightarrow$$2x + 5 < 3$$\Rightarrow$ x < -1 5 &$2x + 5 > 0$$\Rightarrow$ x > - 2 Domain is x$\ in $$(- 5$2 , -1) 5$\gamma$= - ,$\delta$= -1 2 | 2 5 | |7($\alpha$+$\beta$) + 4($\gamma$+$\delta$)$| = |$7$( -12$+ + 4 (- - 1)| | 7 2 |$| - 82 - 14| = 96$
Correct Answer: 96