Problem 30 - Precalculus with Calculus Previews

Question

In Problems \(25-32\), find the domain of the given function \(f .\) Find the \(x\) -intercept and the vertical asymptote of the graph. Use transformations to graph the given function \(f\). $$ f(x)=1-2 \log _{4}(x-4) $$

Step-by-Step Solution

Answer

Domain: \((4, \infty)\), x-intercept: \((6, 0)\), Vertical asymptote: \(x = 4\).

Step 1: Determine the Domain

The domain of a logarithmic function \(\log_b(x-a)\) consists of all \(x\) values for which the argument \(x-a > 0\). For our function \(f(x) = 1 - 2\log_4(x-4)\), the argument is \(x-4\). Thus, we need \(x-4 > 0\). Solving for \(x\), we get \(x > 4\). Therefore, the domain of \(f(x)\) is \((4, \infty)\).

Step 2: Find the x-intercept

To find the \(x\)-intercept, set \(f(x)\) to zero and solve for \(x\):\[ 0 = 1 - 2\log_4(x-4) \]Rearrange the equation to isolate the logarithm term:\[ 2\log_4(x-4) = 1 \]Divide by 2:\[ \log_4(x-4) = \frac{1}{2} \]Convert the logarithm to exponential form:\[ x-4 = 4^{\frac{1}{2}} \]Since \(4^{\frac{1}{2}} = 2\), add 4 to both sides:\[ x = 4 + 2 = 6 \]Hence, the \(x\)-intercept is at \((6, 0)\).

Step 3: Identify the Vertical Asymptote

The vertical asymptote occurs where the logarithm becomes undefined, which is when the argument equals zero. In this case, at \(x - 4 = 0\), so \(x = 4\). Hence, there is a vertical asymptote at \(x = 4\).

Step 4: Graph the Function Using Transformations

Start with the basic logarithmic function \(\log_4(x)\). This function has a vertical asymptote at \(x=0\) and passes through \((1,0)\).1. **Shift Right by 4:** Transform \(\log_4(x)\) to \(\log_4(x-4)\), shifting the graph right by 4 units, moving the vertical asymptote to \(x = 4\) and passing through \((5,0)\).2. **Vertical Stretch by 2:** Multiply by -2 to get \(-2\log_4(x-4)\), stretching the graph vertically by a factor of 2. This changes the point \((5,0)\) to \((5,-2)\).3. **Vertical Shift Up by 1:** Finally, add 1, resulting in the function \(1 - 2\log_4(x-4)\), which shifts the entire graph up by 1 unit, shifting the point \((5,-2)\) to \((5,-1)\).Plot the transformed function with the identified intercepts and asymptote.

Key Concepts

Domain of a Function

When working with logarithmic functions like \[f(x) = 1 - 2 \log_4(x-4)\], it's important to determine the domain first. The domain is the set of all possible input values \(x\) for which the function is defined. For a logarithmic function, this is where the argument of the logarithm, in this case \(x-4\), is greater than zero.

• Set the argument \(x-4\) greater than zero: \(x-4 > 0\).
• Solve for \(x\): This gives \(x > 4\).

Therefore, the domain of the function is all real numbers greater than 4. Expressing this in interval notation gives us \((4, \infty)\). This means the function is only defined for all \(x\) values strictly greater than 4.

Vertical Asymptote

The vertical asymptote of a logarithmic function is a vertical line where the function's value approaches infinity or negative infinity. For our function, this happens when the argument of the logarithm equals zero.

• Set \(x-4 = 0\).
• Solving gives \(x = 4\).

So, the vertical asymptote is at \(x = 4\).
This means as \(x\) approaches 4 from the right, the value of the function \(f(x)\) heads towards negative infinity. An asymptote represents a boundary in the graph that the function will nearly touch but never actually reach.

Graph Transformations

To graph a function like \(f(x) = 1 - 2 \log_4(x-4)\), we can apply several graph transformations to the basic logarithm function \(\log_4(x)\). These transformations help us visualize how the function behaves.

• **Shift Right by 4 Units:** Start by shifting \(\log_4(x)\)by 4 units to the right, resulting in \(\log_4(x-4)\). This moves the vertical asymptote from \(x=0\) to \(x=4\).
• **Vertical Stretch by 2:** Multiply by -2 to transform into \(-2\log_4(x-4)\). This stretches the graph vertically by a factor of 2 and flips it upside down because of the negative sign.
• **Vertical Shift Up by 1:** Finally, add 1, resulting in \(1 - 2\log_4(x-4)\). This shifts the entire graph up by one unit, altering the function's baseline level.

Each of these operations transforms the original function in a way that represents its new shape and position on the coordinate plane, making it easier to understand how the function behaves relative to its simpler form. These steps allow you to plot key points and asymptotes accurately.