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The domain of a function refers to all the possible input values (commonly represented as \( x \)) that allow the function to output real values. When dealing with a logarithmic function like \( y = \log (x-2) + 1 \), it's crucial to consider the domain carefully because logarithms are undefined for zero and negative numbers.

For the function \( \log(x-2) +1 \), the argument of the logarithm (which is \( x-2 \) in this case) must be greater than zero. This leads us to the inequality \( x-2 > 0 \). Solving this inequality is simple: just add 2 to both sides, which gives us \( x > 2 \).

This means the domain of our function is all real numbers greater than 2. In mathematical notation, we express this as \( x \in (2, \infty) \).

• Logarithmic functions require their arguments to be positive.
• To determine a domain, solve the inequality that makes the argument positive.
• The domain for our function is \( x > 2 \).

The range of a function is the set of all possible output values. For most logarithmic functions, the range is all real numbers. This is due to the nature of logarithmic growth.

Looking at the function \( y = \log (x-2) + 1 \), let's break it down:

• The part \( \log(x-2) \) can take any real number as a value when its argument \( x-2 \) ranges over positive numbers. When \( x-2 = 1 \), the log value is zero, and as \( x \) approaches 2 from the right, the log value becomes very negative.
• On the other hand, as \( x \) becomes very large, \( \log(x-2) \) increases without bound.

By adding 1 to the entire logarithmic expression, we shift this range vertically by 1 unit. However, this does not change the overall range of the function from all real numbers.

This means the range of our function is all real numbers, expressed as \( y \in \mathbb{R} \). Even though we shifted the function by adding 1, the range remains unchanged.

• Logarithmic ranges are typically all real numbers.
• For \( \log(x-2) + 1 \), the range remains \( y \in \mathbb{R} \).

Inequalities often come into play when determining the domain of a function, particularly for functions like logarithms and square roots, which have restrictions based on their mathematical properties.

In the function \( y = \log(x-2) + 1 \), the fundamental inequality arises from the nature of logarithms. The expression inside the logarithm, \( x-2 \), must remain positive—that is, greater than zero—because the logarithm of zero or a negative number is undefined. This results in the inequality \( x-2 > 0 \), which simplifies to \( x > 2 \).

Understanding how to solve inequalities like this is a key skill in working with functions. Here are a few key points:

• Identify the part of the function that might cause restrictions (e.g., logarithms and their arguments).
• Set up inequalities based on those restrictions (e.g., argument > 0).
• Solve the inequalities to find the permissible values for the domain.

Using these steps can help ensure that you correctly determine when a function is defined and how its domain might be limited due to an inequality.