Problem 194 - Precalculus

Question

For the following exercises, state the domain and range of the function. $$f(x)=\log _{3}(x+4)$$

Step-by-Step Solution

Answer

Domain: \((-4, \infty)\); Range: \((-\infty, \infty)\).

Step 1: Identify the Function Type

First, recognize that the given function is a logarithmic function. In particular, it is a logarithm with base 3, written as \( f(x) = \log_3(x + 4) \).

Step 2: Determine the Domain of the Function

The logarithmic function \( \log_b(x) \) is defined only for positive values of \( x \). So, we need \( x + 4 > 0 \). By solving \( x + 4 > 0 \), we find that \( x > -4 \). Hence, the domain of the function is all real numbers greater than \(-4\), which is \( (-4, \infty) \).

Step 3: Determine the Range of the Function

For any logarithmic function \( \log_b(x) \), where \( b > 1 \), the range is all real numbers. Since our function is \( f(x) = \log_3(x + 4) \), the range of the function is \( (-\infty, \infty) \).

Step 4: State the Domain and Range

Combine the results from Steps 2 and 3. The domain of \( f(x) = \log_3(x + 4) \) is \( (-4, \infty) \), and the range is \( (-\infty, \infty) \).

Key Concepts

Logarithmic Functions

Logarithmic functions are a special type of mathematical function closely related to exponential functions. One key aspect of these functions is the presence of a base, denoted as the subscript in a logarithm. For instance, in the function \( f(x) = \log_3(x + 4) \), the base is 3. The base determines how the function stretches or compresses along the x-axis.
Logarithms help in solving exponential equations and are widely used across scientific fields due to their ability to model growth processes like population growth and radioactive decay. When graphed, a basic logarithmic function passes through the point \((1,0)\), as any logarithm with a positive base to the power of zero equals one.
Key properties of logarithmic functions include:

• They are continuously increasing if the base is greater than one.
• They have a vertical asymptote along the y-axis, indicating that the function values approach negative infinity as the input nears zero from the positive side.

Understanding these characteristics allows us to better comprehend their behavior and apply them to real-world problems.

Domain and Range

When discussing functions, the domain refers to all possible input values (x-values) that a function can accept without leading to undefined results. For the function \( f(x) = \log_3(x + 4) \), the input value \( x+4 \) must be greater than zero because logarithms are only defined for positive numbers. Solving \( x + 4 > 0 \) simplifies to \( x > -4 \), thus giving us the domain of \((-4, \infty)\).
In contrast, the range of a function is the set of possible output values (y-values). For a standard logarithmic function with a base greater than one, such as the one given, the output ranges over all real numbers. Consequently, the range of \( f(x) = \log_3(x + 4) \) is \((-\infty, \infty)\).
These two concepts are essential for understanding how a function behaves, helping us determine where the function exists on a graph and the values it can produce. Having a firm grasp of domain and range allows students to more effectively interpret and use functions in various mathematical contexts.

Real Numbers

Real numbers are a fundamental part of mathematics, encompassing all possible numerical values on the number line, including both rational and irrational numbers. Rational numbers can be expressed as fractions, like \( \frac{1}{2} \), while irrational numbers cannot be represented as finite decimals or fractions, such as \( \pi \) or \( \sqrt{2} \).
The significance of real numbers in functions like \( f(x) = \log_3(x + 4) \) lies in how they are used to determine the domain and range. For the domain, real numbers greater than \(-4\) ensure that the expression inside the logarithm remains positive. The range, being all real numbers, indicates that the function can produce any real number as an output.
Understanding real numbers and their properties is crucial for solving precalculus problems, as these numbers form the backbone of mathematical analysis, calculus, and advanced computations.