Question

In Exercises \(7-20,\) find the vertical asymptotes (if any) of the function.]$$ f(x)=\ln (x+3) $$

Short answer

The function \(f(x)=\ln(x+3)\) has a vertical asymptote at \(x=-3\).

Step-by-step solution

Understand the Function

The given function is \(f(x)=\ln(x+3)\), which is a logarithmic function with a base \(e\). This function shifts the standard logarithmic function \(\ln(x)\) to the left by 3 units.

Identify the Domain

The domain of a logarithmic function \(\ln(x)\) for \(x>0\), or in other words, the function only exists when the argument of the logarithm is positive. In this case, it implies \(x+3 > 0\). Solving this gives us \(x>-3\).

Recognize the Asymptote

In case of logarithmic functions, the function will approach negative infinity when argument of logarithm approaches zero. As per our domain, \(x+3=0\) gives \(x=-3\). So, \(f(x)\) becomes undefined and reaches negative infinity when x approaches -3 from the right. This makes \(x=-3\) the vertical asymptote of the function.

Key concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and have the form \( f(x) = \log_b(x) \) where \( b \) is the base and \( x \) is the argument. The base \( e \)—Euler's number—plays a special role in mathematics and gives rise to the natural logarithm, written as \( \ln(x) \).

When discussing logarithms, it's crucial to note that they are only defined for positive arguments. This is because asking for the logarithm of a negative or zero value does not make sense within real numbers—a logarithm essentially asks, 'To what power must we raise \( b \) to get \( x \)?', and \( b \) raised to any real power cannot yield a negative number or zero.

Adjustments in the argument, as seen in the function \( f(x)=\ln(x+3) \) from the exercise, result in shifting the basic \( \ln(x) \) graph horizontally. Such transformations are important as they affect the domain and asymptotic behavior of the function.

Domain of a Function

The domain of a function consists of all the input values (x-values) for which the function is defined and yields a real number output. Determining the domain is a fundamental step in graphing and analyzing functions.

For logarithmic functions like \( f(x)=\ln(x+3) \) from our exercise, the domain is restricted to the set of \( x \) values that keep the argument of the logarithm positive. This is due to the nature of a logarithm which does not accept zero or negative numbers as input. Therefore, identifying the domain involves setting \( x+3 > 0 \) and solving for \( x \) to find \( x>-3 \) as the valid domain. This pivotal step informs us about the function's behavior and leads to understanding its asymptotic properties.

Asymptotic Behavior

Asymptotic behavior describes how a function behaves as it gets very close to a certain line or point, usually at extreme values of \( x \). A vertical asymptote of a function is a vertical line \( x=a \) where the function tends to go towards positive or negative infinity as it approaches \( a \) from the left or right.

In the case of the given logarithmic function \( f(x)=\ln(x+3) \) we find that as \( x \) approaches -3 from the right, the argument of the logarithm approaches zero, and the function's output heads towards negative infinity. This defines \( x=-3 \) as a vertical asymptote. This asymptotic behavior is crucial to understand since it affects how the graph of the function is drawn and how the function behaves around those critical values.