Problem 233 - Precalculus

Question

For the following exercises, sketch the graph of the indicated function. $$g(x)=\log (6-3 x)+1$$

Step-by-Step Solution

Answer

The graph of \(g(x) = \log(6 - 3x) + 1\) has a vertical asymptote at \(x = 2\) and shifts up by 1 unit.

Step 1: Identify the Base Function

The given function is \(g(x) = \log(6 - 3x) + 1\). We recognize that this is a logarithmic function based on the natural logarithm \( \log(x) \). The base function here is \( \log(x) \), which is typically logarithm to the base 10.

Step 2: Determine the Domain

Since \( g(x) = \log(6 - 3x) + 1 \), the argument of the logarithm \( 6 - 3x \) must be positive. Therefore, solve the inequality \( 6 - 3x > 0 \), which simplifies to \( x < 2 \). Thus, the domain of \( g(x) \) is all real numbers \( x \) such that \( x < 2 \).

Step 3: Find the Vertical Asymptote

For the function \( \log(6 - 3x) + 1 \), a vertical asymptote occurs where the argument of the logarithm is zero. Setting \( 6 - 3x = 0 \), we solve for \( x \) and find that \( x = 2 \). This means there is a vertical asymptote at \( x = 2 \).

Step 4: Determine the Horizontal Transformation

The expression \( 6 - 3x \) suggests a horizontal shift and a horizontal scaling. The factor \( -3 \) indicates a reflection over the y-axis and compression, as well as affecting the direction of the input for the logarithm (reversing interval direction). The term '6' in the argument \( 6 - 3x \) shifts the graph horizontally to the left.

Step 5: Vertical Shift

The function has a \(+1\) outside of the logarithmic function, indicating a vertical upward shift of 1 unit for \( \log(6 - 3x) \). This moves every point on the graph up by 1 unit.

Step 6: Sketch the Graph

To sketch the graph, start with the vertical asymptote at \( x = 2 \). Since the graph of a logarithm generally increases left to right, near the asymptote \( x < 2 \), the function will tend towards negative infinity and rise as \( x \) decreases. Shift the entire graph up by 1 unit due to the vertical shift and reflect the usual increasing behavior due to the factor \(-3\). Plot several points to aid the sketch, such as where \( x = 0 \), \( g(0) = \log(6) + 1 \).

Key Concepts

Domain of a Function

In mathematics, the domain of a function is the set of all possible input values (or 'x' values) that a function can accept without resulting in any mathematical issues. For the given function, \( g(x) = \log(6 - 3x) + 1 \), we need to consider when the logarithmic function is defined. This is because a logarithm requires its argument to be a positive number.

Thus, we solve the inequality:

• \( 6 - 3x > 0 \)
• Solve for \( x \): \( 6 > 3x \)
• Divide both sides by 3: \( 2 > x \) or \( x < 2 \)

So, the domain of \( g(x) \) consists of all real numbers less than 2. This means the function accepts any \( x \) value smaller than 2, keeping the logarithmic expression always positive and valid.

Vertical Asymptote

A vertical asymptote in the graph of a function is a vertical line where the function approaches infinity or negative infinity as the input value nears a specific point. It marks the boundaries where the function does not exist. For the function \( g(x) = \log(6 - 3x) + 1 \), the vertical asymptote arises when the argument of the logarithm becomes zero.

In other words:

• Set \( 6 - 3x = 0 \)
• Solve for \( x \): \( 3x = 6 \)
• Thus, \( x = 2 \)

Hence, you will find a vertical asymptote at \( x = 2 \). As \( x \) approaches this value from the left, \( g(x) \) reaches negative infinity because you can't take the logarithm of zero or negative numbers. This is key in graphing the function accurately.

Horizontal Transformations

Horizontal transformations involve shifts and reflections that affect how a graph is positioned and oriented on the Cartesian plane. For the function \( g(x) = \log(6 - 3x) + 1 \), look closely at the term \( 6 - 3x \) inside the logarithm.

Here’s what happens:

• The \( -3 \) factor indicates reflection about the y-axis.
• This factor also causes a horizontal compression. This means that the function will change more quickly than \( \log(x) \) alone.
• The \( 6 \) shifts the graph horizontally to the left.

Remember, horizontal transformations can be counter-intuitive:

• Negative inside arguments like \( -3 \) reverse the direction.
• A positive constant like \( +6 \) shifts the graph left, rather than right.

These transformations reshape the standard logarithmic curve across the horizontal axis, making them crucial in graph sketching.

Graph Sketching

Graph sketching involves translating the mathematical understanding of a function into a visual representation. For \( g(x) = \log(6 - 3x) + 1 \), here's how to graph it:

• Start by adding the vertical asymptote at \( x = 2 \).
• Since the base function \( \log(x) \) has a characteristic shape that climbs steadily, the graph will rise to the left of this asymptote.
• Adjust for a vertical shift: Add 1 to every point on the logarithmic graph.
• Consider the reflection and compression by the factor \(-3\); this causes the function to decrease more rapidly and appears reversed.

Include a few points to clarify the shape:

• For \( x = 0 \), calculate \( g(0) = \log(6) + 1 \).
• Note how graph's direction shifts due to transformations.

Sketching this function requires a firm grasp of all transformations, ensuring that your graph accurately reflects the mathematical structure of the function.