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What is the domain of the function [tex]$y=\ln \left(\frac{-x+3}{2}\right)$[/tex]?

A. [tex]$x\ \textless \ 2$[/tex]
B. [tex][tex]$x\ \textgreater \ 2$[/tex][/tex]
C. [tex]$x\ \textless \ 3$[/tex]
D. [tex]$x\ \textgreater \ 3$[/tex]


Sagot :

To determine the domain of the function [tex]\( y = \ln \left( \frac{-x + 3}{2} \right) \)[/tex], we must ensure that the argument of the logarithmic function is positive, because the natural logarithm, [tex]\( \ln(z) \)[/tex], is defined only for [tex]\( z > 0 \)[/tex].

Let's start by setting up the inequality:

[tex]\[ \frac{-x + 3}{2} > 0 \][/tex]

To solve this inequality:

1. Multiply both sides of the inequality by 2 to clear the fraction:

[tex]\[ -x + 3 > 0 \][/tex]

2. Next, solve for [tex]\( x \)[/tex] by isolating it on one side of the inequality:

[tex]\[ -x > -3 \][/tex]

3. Divide both sides by -1, and remember to reverse the inequality sign when dividing by a negative number:

[tex]\[ x < 3 \][/tex]

Thus, the domain of the function [tex]\( y = \ln \left( \frac{-x + 3}{2} \right) \)[/tex] is all [tex]\( x \)[/tex] such that [tex]\( x < 3 \)[/tex].

Therefore, the correct answer is:

[tex]\[ x < 3 \][/tex]