91Ó°ÊÓ

When exploring the function \( f(x) = \log(a-x) - b \), the domain and range are key aspects to understand its behavior. The domain refers to the set of all possible input values (\(x\)-values) for which the function is defined. For a logarithmic function like \( \log(a-x) \), the expression is only valid if the argument inside the log is greater than zero. Therefore, we require \( a-x > 0 \), which simplifies to \( x < a \). This means the domain of \( f(x) \) consists of all real numbers less than \( a \). In interval notation, this domain is expressed as \((-\infty, a)\).

On the other hand, the range of a function deals with all possible output values (or \(y\)-values) that the function can produce. For \( \log(a-x) \), the outputs can extend from negative infinity to positive infinity. When we subtract \( b \) from \( \log(a-x) \), the entire set of outputs is simply shifted downward by \( b \) units. However, the essential infinity to infinity nature of the possible outputs remains unaffected. Consequently, the range of \( f(x) \) remains all real numbers, \((-\infty, \infty)\).

The \(x\)-intercept is the point on the graph where the function \( f(x) = \log(a-x) - b \) crosses the \(x\)-axis. This happens when the output or \(y\)-coordinate is zero, meaning \( f(x) = 0 \). To find this intercept, we set the function equal to zero:

• \( \log(a-x) - b = 0 \).
• Add \( b \) to both sides to isolate the logarithm: \( \log(a-x) = b \).
• To solve for \( x \), we convert the logarithmic equation to its exponential form. The equation \( \log(a-x) = b \) implies \( a-x = e^b \).
• Solve for \( x \) to find \( x = a - e^b \).

Thus, the \( x \)-intercept is at the point \((a - e^b, 0)\), meaning this is where the function’s graph touches the \(x\)-axis.

Logarithmic inequalities are fundamental in defining the behavior of functions like \( f(x) = \log(a-x) - b \). They help set constraints for what \(x\) can be, ensuring that the function and its operations stay valid and well-defined. In this case, the inequality \( a-x > 0 \) originates from the property of logarithms that only positive values can be input. This specific inequality simplifies to \( x < a \), outlining the domain bound.

In general, solving logarithmic inequalities follows a systematic process:

• Convert the logarithmic inequality into a more familiar algebraic form by exponentiating both sides if necessary.
• Consider the base of the logarithm. For instance, if the base were 10, the logarithm could be expressed as \(10^y = a-x\), where \(y\) is the right side of an inequality being solved.
• Analyze the transformed inequality to find the \(x\)-values that satisfy the requirement. Depending on whether the base is greater or less than one, rules for flipping inequalities might apply.

This step is essential for maintaining mathematical accuracy, ensuring all solutions are real numbers conforming to the function's established domain.