Problem 18 - College Algebra

Question

For the following exercises, state the domain, vertical asymptote, and end behavior of the function. $$h(x)=-\log (3 x-4)+3$$

Step-by-Step Solution

Answer

Domain: \((\frac{4}{3}, \infty)\); Vertical Asymptote: \(x = \frac{4}{3}\); End Behavior: \(h(x) \to +\infty\) as \(x \to \frac{4}{3}^+\), \(h(x) \to -\infty\) as \(x \to \infty\).

Step 1: Find the Domain

The domain of the function is determined by the logarithm's argument, which must be greater than zero. Therefore, we set \(3x - 4 > 0\). Solving for \(x\), we add 4 to both sides to get \(3x > 4\), then divide both sides by 3, yielding \(x > \frac{4}{3}\). Thus, the domain is \(x \in (\frac{4}{3}, \infty)\).

Step 2: Identify the Vertical Asymptote

A vertical asymptote occurs at the values of \(x\) that result in the logarithm's argument being zero. From the inequality \(3x - 4 > 0\), we find that \(3x - 4 = 0\) when \(x = \frac{4}{3}\). Therefore, there is a vertical asymptote at \(x = \frac{4}{3}\).

Step 3: Analyze the End Behavior

The end behavior of the function is considered as \(x\) approaches the extremes of its domain. As \(x \to \frac{4}{3}^+\), \(3x - 4\) approaches 0 from the right, making \(\log(3x - 4)\) approach \(-\infty\). Consequently, \(-\log(3x - 4)\) approaches \(+\infty\), and adding 3 does not change the infinity. Therefore, \(h(x) \to +\infty\) as \(x \to \frac{4}{3}^+\). As \(x \to \infty\), \(3x - 4\) results in the logarithm expression growing large, making \(\log(3x - 4)\) approach \(+\infty\). Thus, \(-\log(3x - 4)\) approaches \(-\infty\), and \(h(x) \to -\infty\).

Key Concepts

Domain of a Function

The domain of a function is the set of all possible input values, usually represented by "x", for which the function is defined. When dealing with logarithmic functions, special attention must be paid to the argument of the logarithm.
Logarithms are only defined for positive numbers, meaning that the argument inside the \(\log\) must be greater than zero.
For the function \(h(x) = -\log (3x-4) + 3\), we start by considering the expression \(3x-4\).
We need to ensure this expression is positive:

• \(3x - 4 > 0\)
• Solving the inequality gives \(x > \frac{4}{3}\)

.
Therefore, the domain of the function \(h(x)\) is \(x \in \left(\frac{4}{3}, \infty \right)\). This means the function is defined for all values of \(x\) that are greater than \(\frac{4}{3}\). It is crucial to always check that the argument of the logarithm is positive when determining the domain.

Vertical Asymptote

Vertical asymptotes are lines that the graph of a function approaches but never quite reaches. They commonly arise in rational and logarithmic functions. For logarithmic functions, vertical asymptotes typically occur where the argument of the logarithm is zero because the logarithm of zero is undefined.
In the function \(h(x) = -\log (3x-4) + 3\), we find the vertical asymptote by setting the logarithm's argument equal to zero:

• \(3x - 4 = 0\)
• Solve for \(x\) to find \(x = \frac{4}{3}\)

. This method tells us that there is a vertical asymptote at \(x = \frac{4}{3}\). This is the point where the function does not exist and the graph tends to infinity (either positive or negative) as it gets close to \(x = \frac{4}{3}\). Remember, vertical asymptotes signify that as \(x\) approaches this value from either side, the function rises or falls without bound.

End Behavior

The end behavior of a function describes how the function behaves as the input values become very large or very small. For logarithmic functions like \(h(x) = -\log(3x-4) + 3\), we assess how the function behaves as \(x\) approaches the edges of the domain.
- As \(x \to \frac{4}{3}^{+}\): - The expression inside the log \(3x-4\) gets very close to zero, but from the positive side. - Therefore, \(\log(3x-4) \to -\infty\), turning \(-\log(3x-4) \to +\infty\). - Consequently, \(h(x) \to +\infty\).- As \(x \to \infty\): - \(3x - 4\) becomes very large, and so does the logarithm. - \(\log(3x-4) \to +\infty\), making \(-\log(3x-4) \to -\infty\). - Thus, \(h(x) \to -\infty\).The end behavior analysis helps you understand how the function "takes off" towards infinity near its vertical asymptote, and dives downwards as \(x\) increases indefinitely.