Problem 206 - Precalculus

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

Step 1: Analyze the Inverse Logarithmic Function

The function given is in the form \( h(x) = -\log (3x - 4) + 3 \). The function \( y = \log(a) \) is defined when \( a > 0 \). Here, \( a \) is \( 3x - 4 \), so \( 3x - 4 > 0 \).

Step 2: Solve for the Domain

To find the domain of \( h(x) \), solve the inequality: \( 3x - 4 > 0 \). Add 4 to both sides: \( 3x > 4 \). Divide by 3: \( x > \frac{4}{3} \).Thus, the domain of \( h(x) \) is \( x \in \Bigg( \frac{4}{3}, \infty \Bigg) \).

Step 3: Identify the Vertical Asymptote

A vertical asymptote occurs when the expression inside the logarithm equals zero, leading to \(-\infty\). For \( h(x) \), set \( 3x - 4 = 0 \). Solving gives: \( 3x = 4 \), so \( x = \frac{4}{3} \).Therefore, the vertical asymptote is at \( x = \frac{4}{3} \).

Step 4: Determine End Behavior

Determine the behavior of \( h(x) \) as \( x \to \infty \). As \( x \to \infty \), the term \( 3x - 4 \) also goes to \( \infty \), making \( \log(3x - 4) \to \infty \). Hence, \(-\log(3x - 4) \to -\infty\), thus \( h(x) \to -\infty \) as well. Thus, as \( x \to \infty \), \( h(x) \to -\infty \).

Key Concepts

Domain of a Function

The domain of a function refers to the complete set of possible input values (or 'x' values) that the function can accept. For a logarithmic function like \( h(x) = -\log (3x - 4) + 3 \), it's crucial to know when the expression inside the logarithm is positive because the logarithm is only defined for positive values.
Let's break it down:- The expression inside the log is \( 3x - 4 \).- We want \( 3x - 4 > 0 \) for the function to be defined. - Solving the inequality \( 3x - 4 > 0 \), we add 4 to both sides, resulting in \( 3x > 4 \).- Next, divide both sides by 3 to isolate \( x \), yielding \( x > \frac{4}{3} \). Consequently, the domain of \( h(x) \) is all values of \( x \) that are greater than \( \frac{4}{3} \), denoted as \( x \in \left( \frac{4}{3}, \infty \right) \). This means \( h(x) \) is only defined for inputs that satisfy this inequality, ensuring the logarithm's argument is positive.

Vertical Asymptote

A vertical asymptote is a line on the graph of a function where the function approaches infinity or negative infinity. In simpler terms, it's where the function "shoots up" or "dives down" abruptly and is not defined at a particular \( x \)-value.
To find the vertical asymptote for \( h(x) = -\log (3x - 4) + 3 \), we set the expression inside the logarithm to zero, because this will cause the logarithm itself to become undefined and the function to reach negative infinity:
- Solve \( 3x - 4 = 0 \).- Add 4 to both sides, yielding \( 3x = 4 \).- Divide both sides by 3 to isolate \( x \), thus \( x = \frac{4}{3} \).Therefore, the vertical asymptote for the function \( h(x) \) is at \( x = \frac{4}{3} \). This means that as \( x \) approaches \( \frac{4}{3} \) from the right, \( h(x) \) will drop down toward negative infinity.

End Behavior

The end behavior of a function describes what happens to the function's value as \( x \) becomes very large (approaching infinity) or very small (approaching negative infinity). For many functions, understanding end behavior helps to describe how the graph behaves towards its extremities.
Examining \( h(x) = -\log(3x - 4) + 3 \), we consider its behavior as \( x \to \infty \):- As \( x \) grows larger, \( 3x - 4 \) also increases without bounds.- This increase causes \( \log(3x - 4) \) to approach infinity because the logarithmic function grows with its input.- Since our function is \( -\log(3x - 4) + 3 \), the negative sign in front of the logarithm causes the whole expression \( -\log(3x - 4) \) to approach \(-\infty\).Therefore, as \( x \to \infty \), \( h(x) \to -\infty \). The end behavior on this function indicates that as you move further to the right along the x-axis, the value of the function steadily decreases without bound.