Problem 66 - Precalculus

Question

Sketch the graph of \(y=g(x)\) by starting with the graph of \(y=f(x)\) and using transformations. Track at least three points of your choice and the vertical asymptote through the transformations. State the domain and range of \(g\). \(f(x)=\log _{3}(x), g(x)=-\log _{3}(x-2)\)

Step-by-Step Solution

Answer

The graph of \(g(x)\) is a reflection of \(f(x)\) shifted right by 2 units. Domain: (2, ∞); Range: (-∞, 0).

Step 1: Identify the Base Graph

Start with the graph of the basic function, \(f(x) = \log_{3}(x)\). This is a logarithmic function with a vertical asymptote at \(x=0\), and typically passes through the points (1,0), (3,1), and (\frac{1}{3}, -1). The domain of this function is \((0, \infty)\), and the range is \((-\infty, \infty)\).

Step 2: Apply Horizontal Translation

Transform \(f(x)\) horizontally to create \(y = \log_{3}(x-2)\). This shifts the graph of \(y=\log_{3}(x)\) by 2 units to the right. The new vertical asymptote is at \(x=2\), and the tracked points now are: (3,0), (5,1), and (\frac{7}{3}, -1).

Step 3: Apply Vertical Reflection

Apply a vertical reflection to form \(y = -\log_{3}(x-2)\). This reflects the graph over the x-axis, turning it upside-down. Fix each of the tracked points: (3,0) remains the same as it is on the axis, (5,1) turns into (5,-1), and (\frac{7}{3},-1) becomes (\frac{7}{3},1).

Step 4: Determine Domain and Range

Consider the transformations made. The horizontal shift does not affect the range but adjusts the domain. The resulting function \(g(x)\) has a domain of \((2, \infty)\) due to the rightward shift and still a range of \(( -\infty, 0)\) because it has been flipped vertically.

Key Concepts

Logarithmic Functions

Logarithmic functions are a type of function that involve the logarithm, typically with a specific base. In this case, the function is given by

• \(f(x) = \log_{3}(x)\)

The base of the logarithm indicates how the logarithmic scale is set. For a base of 3, every increase by a factor of 3 in the input scales the output linearly. This function has crucial features, such as:

• A vertical asymptote at \(x = 0\)
• Passes through the point \((1,0)\), meaning when \(x = 1\), the output value is 0, following the rule that the logarithm of 1 is 0 in any base

Logarithmic functions are often used to model growth processes, such as population growth, which appear to be linear when plotted on a logarithmic scale. They serve as inverses to exponential functions, compressing the wide range of values that exponentials can take into a more manageable form.

Domain and Range

The domain and range are foundational concepts that determine the set of input (domain) and output (range) values of a function. For any logarithmic function like

• \(f(x) = \log_{3}(x)\)

The domain is the set of all positive real numbers

• \((0, \infty)\) because you can't take the logarithm of zero or a negative number.

The range is all real numbers,

• \((-\infty, \infty)\)

Reflecting or shifting this graph modifies the domain and range of the new function. For the transformed function

• \(g(x) = -\log_{3}(x-2)\)

The domain changes to

• \((2, \infty)\) due to a horizontal shift to the right.

However, the range remains unchanged unless the sign of the output is modified, as seen in vertical reflections.

Horizontal Translation

Horizontal translation in functions involves shifting the entire graph of a function left or right. The mathematical expression for a horizontal translation of the base logarithmic function,

• \(f(x) = \log_{3}(x)\)

by 2 units to the right is

• \(y = \log_{3}(x-2)\)

This transformation is simply adjusting the input value, subtracting 2 from \(x\), pushing the entire curve so that the vertical asymptote moves from

• \(x = 0\) to \(x = 2\)

Effectively, horizontal translations do not alter the function's range but change where the domain starts. So, the vertical asymptote shifts accordingly, and any key points, such as (1,0), move to new positions, such as (3,0), reflecting the shift rightward.

Vertical Reflection

Vertical reflection takes a graph and flips it over the x-axis. For logarithmic functions, this means the output values are inverted. For example, if the original function is

• \(y = \log_{3}(x-2)\)

Applying a vertical reflection results in:

• \(y = -\log_{3}(x-2)\)

Each point on the function that had a positive y-coordinate now has a negative y-coordinate, and vice versa. This conversion is quite visually intuitive if you think about flipping a U-shaped curve upside-down. Additionally, while reflection across the x-axis does not affect the domain, it alters the range by switching all y-values from positive to negative, or vice versa, as the case may be, thus mapping the output set to

• \(( -\infty, 0)\)

for the reflected function.