Problem 68 - Precalculus : Building Concepts and Connections

Question

Find the domain of each function. Use your answer to help you graph the function, and label all asymptotes. $$f(x)=-\log _{2}(x+3)$$

Step-by-Step Solution

Answer

The domain of the function \(f(x)=-\log _{2}(x+3)\) is \(x>-3\). The graph is a downward-opening curve starting above the vertical line \(x = -3\), this line is the vertical asymptote.

Step 1: Find the Domain

The domain of a function is the set of all possible input values which produce valid output. The logarithm function is only defined for positive real numbers. Therefore the expression inside the logarithm, \(x+3\), must be greater than 0, as a result, the domain of the function \(f(x)=-\log _{2}(x+3)\) is \(x>-3\).

Step 2: Identify Asymptotes

Asymptotes are lines to which a curve gets closer but never touches. For a logarithmic function, the vertical asymptote can be found at the boundary of the domain. Here the boundary of the domain is \(-3\), so there is a vertical asymptote at \(x = -3\).

Step 3: Graph the Function

Since this is a negative logarithmic function, the graph will be a downward opening curve. Graph starts from just above the line \(x = -3\) (asymptote), and gradually curves downward to the right. It doesn't touch or cross the line \(x=-3\)

Step 4: Label Asymptotes

Finally, label the asymptote on the graph at \(x = -3\).

Key Concepts

Domain of a Function

Understanding the domain of a function is fundamental in mathematics, particularly when graphing functions. For a given function, the domain refers to all possible values of the independent variable, usually denoted as 'x', for which the function is defined.
When dealing with logarithmic functions like \( f(x) = -\log_2(x+3) \), the domain is restricted to values that make the argument of the logarithm positive since the logarithm of a non-positive number is undefined. Hence, to find the domain, we set the inside of the logarithm greater than zero, resulting in \( x > -3 \).
Exercise Improvement Advice: Emphasize that the domain is all the values that 'x' can take without causing an undefined or non-real result in output. Highlighting and understanding this allows students to avoid common mistakes when sketching the graph of a function.

Vertical Asymptotes

Vertical asymptotes are lines that a function approaches but never touches or crosses. These represent the boundaries beyond which the function doesn’t exist and they commonly occur in rational and logarithmic functions.
In the context of logarithmic functions such as \( f(x) = -\log_2(x+3) \), the vertical asymptote is found at the x-value that makes the argument of the logarithm equal to zero, as the function would approach negative or positive infinity. So for \( f(x) \), the vertical asymptote is the line \( x = -3 \).

Key to Graphing

When graphing a function, identifying vertical asymptotes is vital because it informs the shape and behavior of the graph. It's the barrier that the curve endlessly approaches. For homework or exams, it is essential to label these asymptotes clearly.

Graphing Logarithmic Functions

Graphing logarithmic functions can be a challenge but knowing the rules can simplify the process. The graph of a logarithmic function typically has a characteristic curve that starts from the vertical asymptote and extends into the valid domain of the function.
To graph \( f(x) = -\log_2(x+3) \), start by plotting the vertical asymptote at \( x = -3 \). Since the function includes a negative sign, the graph will be a reflection of the standard log function over the x-axis, meaning it will open downwards. From just above this asymptote, draw a curve that stretches rightward and downward forever, getting closer and closer to the asymptote but never quite touching it.
Exercise Improvement Advice: Encourage students to first plot key points and the asymptote, then sketch the curve accordingly. Remind them that, unlike lines, logarithmic graphs are not straight and require a smooth, continuous curve reflecting the function's properties.