If the domain of the function$2x-3 -1$$4+3x$f$(x) = log$($) + sin$( ) is [$\alpha$,$\beta$) e$5+4x$$2-x$then$\alpha$+ 4$\beta$is equal to 2
Step-by-Step Solution
Key Concept: Impose every domain restriction from logarithms, roots, denominators, and inverse functions before intersecting intervals.
Given function is (2) f$(x) = log$( e$2x-3$$) + sin -1$($4+3x$)$5+4x$$2-x$For domain, the conditions are$2x-3$$4+3x$|$| > 0$and$\le$1$5+4x$|$2-x$| Now,$2x-3$5$3 > 0$$\Rightarrow$ x$\ in $(-$\infty$, - )$\cup$[ ,$\infty$)$5+4x$4 2$4+3x$$and -1$$\le$$2-x$$\le$1$4+3x$$4+3x$$\Rightarrow$$(-1$$\le$)$\cap$($\le$1)$2-x$$2-x$$6+2x$$2+4x$$\Rightarrow$ ($\ge$0)$\cap$($\le$0)$2-x$$2-x$$6 + 2x$$2 + 4x$$\Rightarrow$$\cdot$$\le$0$2 - x$$2 - x$1 $\Rightarrow$ x$\ in $[-3, - ] 2 Hence, we get the domain of f as x$\ in $[-3, - 5 4 ) This means that$\alpha$= -3,$\beta$= - Thus,$\alpha$+ 4$\beta$=$9 - 5 = 4$5 4 2
Correct Answer: 2