Problem 565 - Precalculus

Question

Graph the function \(g(x)=\log (12-6 x)+3\).

Step-by-Step Solution

Answer

The domain of \(g(x)\) is \((-\infty, 2)\), with a vertical asymptote at \(x=2\). The y-intercept is at \((0, \log(12)+3)\).

Step 1: Understand the Domain

The function \(g(x)=\log(12-6x)+3\) involves a logarithm. The logarithmic function is only defined for positive arguments. Set the inside of the logarithm function to be greater than zero: \(12-6x > 0\). Solving this inequality, divide both sides by 6: \(2-x > 0\), which simplifies to \(x < 2\). Thus, the domain of \(g(x)\) is \((-\infty, 2)\).

Step 2: Identify Vertical Asymptote

Vertical asymptotes occur where the function is undefined, corresponding to where the argument inside the logarithm becomes zero. Setting \(12-6x=0\) gives \(x = 2\). Thus, there is a vertical asymptote at \(x=2\).

Step 3: Find Intercepts

For the y-intercept, set \(x=0\) and solve \(g(x)=\log(12-6 \times 0)+3 = \log(12)+3\). We find \(g(0) = \log(12)+3\), which gives an approximate value of \(3\log(10/3)+3\). For x-intercepts, set \(g(x)=0\), which leads to no real solutions within the domain \(-\infty < x < 2\). Thus, there is no x-intercept.

Step 4: Sketch the Graph

Plot the point of the y-intercept found at \((0, \log(12)+3)\) and draw the vertical asymptote at \(x=2\). As \(x\) approaches 2 from the left, the function approaches negative infinity. As \(x\) approaches negative infinity, \(g(x)\) tends to positive infinity. This helps to understand the behavior of the graph as it moves from left to right.

Key Concepts

Domain of a Function

In mathematics, the \( \text{domain of a function} \) refers to all the input values \((x)\) for which the function is defined. When dealing with logarithmic functions like \( g(x) = \log(12 - 6x) + 3 \), it is crucial to determine where the expression inside the log is positive. This is because the logarithm of zero or a negative number is undefined.
To find the domain of \( g(x) \), solve the inequality \( 12 - 6x > 0 \). By dividing both sides by 6, you get \( 2 - x > 0 \), or simply \( x < 2 \).
Hence, the domain of the function is \(( -\infty, 2 )\). This means you can only use values less than 2 for \( x \) in this function.

• Ensure \( x \) is always less than 2 for the function to hold.
• Use only the range \(( -\infty, 2 )\) when plugging numbers into \( g(x) \).

Vertical Asymptotes

\( \text{Vertical asymptotes} \) are lines where a function's output tends towards positive or negative infinity, indicating a point of unbounded behavior. For logarithmic functions like \( g(x) = \log(12 - 6x) + 3 \), vertical asymptotes occur where the argument of the logarithm equals zero. This is because the function becomes undefined as it approaches that value.
To determine the vertical asymptote, set \( 12 - 6x = 0 \) and solve for \( x \). This leads to \( x = 2 \). Consequently, the vertical asymptote is at \( x = 2 \).
The vertical asymptote at \( x = 2 \) implies:

• The graph will approach this line but never actually touch or cross it.
• As \( x \) approaches 2 from the left, \( g(x) \) tends to negative infinity.

Vertical asymptotes help us visualize how the function behaves near these undefined points, guiding the sketch of the graph.

Intercepts of a Graph

Intercepts are essential for understanding where a graph meets the axes. There are two key types:

• \( \text{Y-intercept:} \) where the graph crosses the y-axis \((x = 0)\).
• \( \text{X-intercept:} \) where the graph crosses the x-axis \((y = 0)\).

For the function \( g(x) = \log(12 - 6x) + 3 \), let's explore these intercepts:
**Y-Intercept**: Substitute \( x = 0 \) into \( g(x) \).
\[g(0) = \log(12) + 3\]
This evaluates to approximately \(\log(12) + 3\), giving the y-intercept point as \((0, \log(12) + 3)\).
**X-Intercept**: Set \( g(x) = 0 \) and solve \( \log(12 - 6x) + 3 = 0 \). Solving this for \( x \) within the domain, there are no real solutions, indicating no x-intercepts. This is due to the constraints of the domain \((-\infty, 2)\), where the function does not cross the x-axis.
Understanding intercepts aids greatly in plotting and analyzing the general position and shape of the graph.