Question 24003: Find the domain of the function
f(x) = log10 (x^-8x+15)
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! f(x) = log10 (x^-8x+15)...THIS SHOULD BE log10 (x^2-8x+15)...I THINK
TO FIND DOMAIN OF X ,WE NEED TO FIND VALUES OF X THAT WILL PERMIT CALCULATION OF THE FUNCTION f(x) IN A MATHEMATICALLY MEANINGFULL OR PERMISSIBLE WAY ...
MATHEMATICALLY SPEAKING,DIVISION BY ZERO OR SQUARE ROOT OF NEGATIVE NUMBER OR LOG ZERO OR LOG NEGATIVE NUMBER HAVE NO MEANING.
HENCE WE SHOULD LOOK FOR AND AVOID SUCH SITUATIONS ...
HERE WE HAVE A LOG OF (x^2-8x+15) FUNCTION.SO (x^2-8x+15)=Y SAY SHOULD BE ONLY POSITIVE.SUCH VALUES OF X WHICH MAKE IT POSITIVE ARE ONLY PERMISSIBLE.HENCE THEY CONSTITUTE THE DOMAIN OF X
LET Y =(x^2-8x+15)>0
=X^2-5X-3X+15>0
=X(X-5)-3(X-5)>0
=(X-3)(X-5)>0...NOW , FOR A PRODUCT O 2 FACTORS TO BE POSITIVE ,EITHER BOTH FACTORS SHOULD BE NEGATIVE OR BOTH FACTORS SHOULD BE POSITIVE.HENCE
CASE 1......(X-3)<0 AND (X-5)<0...THAT IS
X<3 AND X<5 OR IN EFFECT X<3......CASE 1.
OR
CASE 2....(X-3)>0 AND (X-5)>0....THAT IS
X>3 AND X>5....OR INEFFECT X>5...CASE 2
HENCE THE DOMAIN OF X IS ALL X < 3 AND ALL X>5
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