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The domain of a function refers to all the possible input values (x-values) that the function can accept. When dealing with logarithmic functions, it is important to ensure that their arguments are greater than zero. This is because you cannot take the logarithm of a negative number or zero.
For the function given in the exercise, which is \( y = \log_{6}(x+1) \), we must determine the values of \( x \) for which the argument \( x+1 \) is greater than zero. This translates to the inequality \( x + 1 > 0 \). Solving this inequality gives us \( x > -1 \).
Thus, the domain of the function \( y = \log_{6}(x+1) \) is all real numbers greater than \(-1\). This is because, for \( x > -1 \), the expression \( x+1 \) remains positive, which is suitable for the logarithmic function.

The range of a function is all possible output values (y-values) that the function can produce. Logarithmic functions are unique because they can produce any real number as an output.
For the function \( y = \log_{6}(x+1) \), the logarithm can output any real number because it continuously increases or decreases as the input \( x+1 \) changes. This behavior makes the range of such a function equal to all real numbers.
Therefore, the range of the function \( y = \log_{6}(x+1) \) is \( (-\infty, +\infty) \). In this interval, the function can output negative, zero, or positive values based on the value of \( x \). It essentially covers the entire y-axis on a graph.

Interval notation is a concise way of representing a set of numbers, often used to describe domains and ranges of functions. It uses brackets and parentheses to indicate the beginning and end of an interval.
In interval notation, a round parenthesis \( \) represents that an endpoint is not included in the interval, while a square bracket \[ \] means the endpoint is included. For example, \( (a, b) \) describes all numbers between \( a \) and \( b \), excluding \( a \) and \( b \) themselves.
The domain of the function \( y = \log_{6}(x+1) \) was found to be \( x > -1 \). In interval notation, this is expressed as \( (-1, +\infty) \), indicating that all numbers greater than \(-1\) are part of the domain. Likewise, the range of the function \( y = \log_{6}(x+1) \), which is all real numbers, is written as \((-\infty, +\infty)\). This notation efficiently conveys the infinite span of the function's range and tells you quickly about the scope of valid inputs and outputs.