Problem 190 - Precalculus

Question

What type(s) of translation(s), if any, affect the range of a logarithmic function?

Step-by-Step Solution

Answer

Only vertical translations affect the range of a logarithmic function.

Step 1: Understand the Logarithmic Function

Let's first understand the nature of the logarithmic function. A basic logarithmic function is of the form \(f(x) = \log_b(x)\), where \(b > 1\). The range of a logarithmic function like this is all real numbers, \((-\infty, \infty)\).

Step 2: Identify Types of Translations

Translations can be either vertical or horizontal. A vertical translation would involve shifting the function up or down (e.g., \(f(x) = \log_b(x) + c\)), while a horizontal translation involves shifting the graph left or right (e.g., \(f(x) = \log_b(x - d)\)).

Step 3: Analyze Vertical Translations

For vertical translations, the function becomes \(f(x) = \log_b(x) + c\), where \(c\) is a constant. Since adding or subtracting a constant affects the output (\(y\)-values), the range will be altered by this translation, resulting in \((-\infty+c, \infty+c)\).

Step 4: Analyze Horizontal Translations

With horizontal translations, the function takes the form \(f(x) = \log_b(x - d)\). This translation changes the input \(x\) values, affecting the domain but not the range. Hence, the range remains \((-\infty, \infty)\), unaffected by horizontal translations.

Key Concepts

Translations

Translations refer to the shifts in a function's graph, and they are an essential concept in understanding how function graphs change position in the coordinate plane. In the context of logarithmic functions, translations mainly affect the graph's position:

• Vertical Translations: These occur when the graph shifts up or down.
• Horizontal Translations: These entail a shift left or right.

The key thing to grasp is how these translations affect the properties of the function, such as domain and range. While translations can change the output and input of a function, the nature of each type determines which of these is impacted.

Range

The range of a function represents all the possible output values (often the y-values) that the function can produce. For the basic form of a logarithmic function, such as \(f(x) = \log_b(x)\), with base \(b > 1\), the range is all real numbers, symbolically represented as \((-\infty, \infty)\).
Logarithmic functions intrinsically cover all real numbers because their output can take any value depending on the input values. It's important to note, however, that not all translations influence the range. Only specific types of translations, namely vertical translations, will modify this characteristic, since they directly affect the y-values of the function outputs.

Vertical Translations

Vertical translations affect the range of a logarithmic function. This type involves adding or subtracting a constant from the function, as seen in the transformation \(f(x) = \log_b(x) + c\), where \(c\) is a constant. These translations move the entire graph up or down on the y-axis.

• When \(c > 0\), the graph shifts upward, leading to a new range of \((-\infty+c, \infty+c)\).
• When \(c < 0\), the graph shifts downward, moving the range to \((-\infty+c, \infty+c)\).

These changes in the range reflect the alteration of the outputs of the function. Hence, with vertical translations, the entire set of y-values adjusts either up or down depending on the value of \(c\).

Horizontal Translations

Horizontal translations change the graph of a function by shifting it left or right. In a logarithmic function, such as \(f(x) = \log_b(x - d)\), the horizontal translation doesn't alter the range. Instead, it influences the domain since it impacts the x-values.

• When \(d > 0\), the graph shifts to the right.
• When \(d < 0\), the graph shifts to the left.

Despite this, the range of a logarithmic function, \((-\infty, \infty)\), remains unchanged with horizontal translations. This is because these shifts do not affect the output values directly, only where the inputs begin or extend in their reach on the x-axis.