Problem 66 - Precalculus with Limits

Question

Find the domain, \(x\) -intercept, and vertical asymptote of the logarithmic function and sketch its graph. \(h(x)=\ln (x+5)\)

Step-by-Step Solution

Answer

The domain of \(h(x)=\ln (x+5)\) is \(x> -5\) or \((-5, +\infty)\), the x-intercept is -4, and the vertical asymptote is \(x=-5\).

Step 1: Identify the domain

The argument of a logarithmic function is always positive. Therefore, the domain of the function is the set of all real numbers \(x\) such that \(x+5>0\). Solving that inequality gives \(x>-5\). So the domain of \(h(x)\) is \(( -5, +\infty)\).

Step 2: Find the x-intercept

The x-intercept of a function is the x-value where the y-value equals 0. Therefore, we set \(h(x)=0\), then we find \(x\) such that \(\ln(x+5)=0\). This is the same as saying \(x+5 = 1\) (since ln 1 = 0). Solving for \(x\) gives \(x = 1 - 5 =-4\). So, the x-intercept is -4.

Step 3: Identify the vertical asymptote

The vertical asymptote of a logarithmic function is the line \(x=a\) where \(a\) is the number such that \(x+a=0\). For \(h(x)\), the vertical asymptote is the line \(x=-5\).

Step 4: Sketch the graph

Once you have found the domain, x-intercept and vertical asymptote, you can proceed to sketch the graph of the function. The graph of \(h(x)\) will be a curve that starts from the asymptote at \(x=-5\), crosses the x-axis at \(x=-4\), and then increases without bound as \(x\) goes to infinity. A key point to remember is that the graph will never touch or cross the vertical asymptote.

Key Concepts

Domain of a Function

The domain of a function in mathematics refers to the set of possible input values, or "x-values," that a function can accept. When dealing with logarithmic functions like the one in this exercise, it is important to remember that the argument of the logarithm must be positive. For example, in the function \( h(x) = \ln(x + 5) \), we ensure the expression inside the logarithm, i.e., \( x + 5 \), remains greater than zero. This leads us to the inequality \( x + 5 > 0 \). Solving the inequality gives us \( x > -5 \). Thus, the domain of the function is all real numbers greater than \(-5\), expressed as \((-5, \infty)\). This means any number greater than \(-5\) can be plugged into the function, while numbers less than or equal to \(-5\) will not work.

X-Intercept

To find the x-intercept of a function, we determine the x-value at which the function's output, or "y-value," equals zero. This essentially means finding the point where the graph crosses the x-axis. For \( h(x) = \ln(x + 5) \), we set \( h(x) = 0 \). This results in the equation \( \ln(x + 5) = 0 \). Since \( \ln(1) = 0 \), it implies that \( x + 5 = 1 \). Solving for \( x \), we get \( x = 1 - 5 = -4 \). Therefore, the function's x-intercept is at \( x = -4 \). At this point, the graph touches the x-axis and then continues to rise as x increases.

Vertical Asymptote

A vertical asymptote of a logarithmic function is a line that the graph approaches but never actually reaches or crosses. This typically occurs at the value where the logarithm is undefined, i.e., its argument equals zero. For the function \( h(x) = \ln(x + 5) \), the argument \( x + 5 \) becomes zero when \( x = -5 \). Hence, the vertical asymptote is the line \( x = -5 \). As \( x \) gets closer to \(-5\) from the right, the function \( h(x) \) decreases without bound, illustrating the behavior that characterizes a vertical asymptote. It's critical in graphing the function to indicate this boundary that the curve will approach but never intersect.

Graphing Functions

In graphing logarithmic functions, several key details guide the process. With \( h(x) = \ln(x + 5) \), first identify the vertical asymptote at \( x = -5 \) and the x-intercept at \( x = -4 \). Start plotting the function by marking the vertical asymptote, which is a dashed line to signify that the graph cannot touch or cross this boundary.
Then, place a point at the x-intercept \(( -4, 0)\). Begin drawing the curve from close to the asymptote and let it pass through the x-intercept, continuing upward as \( x \) increases.
The graph should rise gently, starting from the vertical asymptote towards the x-axis, illustrating the steady increase typical of logarithmic functions. These functions have a distinctive curve that approaches the asymptote from one side and stretches out infinitely without crossing the x-axis again after the intercept. Practicing these steps aids in mastering the art of graphing logarithmic functions effectively.