What is the domain of the function $y=\ln \left(\frac{-x+3}{2}\right)$ ?
To find the domain of the function \(y = \ln\left(\frac{-x + 3}{2}\right)\), we must identify all real values of \(x\) for which the expression is defined. The natural logarithm function \(f(u) = \ln(u)\) is defined only for arguments \(u\) that are strictly positive, meaning \(u > 0\). In our case, the argument is \(\frac{-x + 3}{2}\). Therefore, we need to solve the inequality:
\[\frac{-x + 3}{2} > 0\]
To solve the inequality \(\frac{-x + 3}{2} > 0\), we first multiply both sides by 2. Since 2 is a positive number, the direction of the inequality remains unchanged:
\[-x + 3 > 0\]
Next, we isolate the term with \(x\). Subtracting 3 from both sides gives:
\[-x > -3\]
Finally, to solve for \(x\), we multiply or divide both sides by -1. Recall that multiplying or dividing an inequality by a negative number reverses the direction of the inequality:
\[x < 3\]
Thus, the function is defined for all values of \(x\) that are strictly less than 3.
The domain of the function \(y = \ln\left(\frac{-x + 3}{2}\right)\) is the set of all real numbers \(x\) such that \(x < 3\). Comparing this result with the provided options, we conclude that the correct choice is \(x < 3\).
The concept of a domain is crucial when analyzing real-world systems, such as designing a bridge or managing population growth. For example, if a model for the growth of a bacterial population involves a logarithmic function, the domain tells us the range of time (e.g., \(t < 3\) hours) for which the model is valid. Knowing the constraints ensures that mathematical predictions remain physically meaningful and do not produce undefined results.