Problem 6 - College Algebra

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: \( x > -4 \); Range: all real numbers.

Step 1: Understanding the Logarithmic Function Domain

The domain of a logarithmic function such as \( f(x) = \log_3(x + 4) \) consists of the set of all input values that make the expression inside the logarithmic function positive. Logarithms are undefined for non-positive numbers.

Step 2: Set the Argument Greater Than Zero

Set the argument of the logarithmic function greater than zero to determine the domain. For \( f(x) = \log_3(x+4) \), this means solving \( x + 4 > 0 \).

Step 3: Solve the Inequality for the Domain

Subtract 4 from both sides of the inequality: \( x + 4 > 0 \) becomes \( x > -4 \). Thus, the domain of \( f(x) \) is all real numbers greater than \(-4\).

Step 4: Understanding the Range of a Logarithmic Function

The range of a logarithmic function such as \( f(x) = \log_3(x + 4) \) is all real numbers. Logarithmic functions can produce any real number as an output as the input increases.

Key Concepts

Domain of a Function

When talking about the domain of a function, we refer to all the possible values that can be plugged into the function to get a valid output. For logarithmic functions, like \( f(x) = \log_3(x + 4) \), the focus is on ensuring that the argument of the logarithm is positive. Logarithms are undefined for zero or negative numbers. Thus, finding the domain is all about identifying the set of all x-values that keep the logarithm's input positive.

In the function \( f(x) = \log_3(x + 4) \), the term \( x + 4 \) must be greater than zero because logarithms require positive inputs. Let's break it down:

• Set up the inequality: \( x + 4 > 0 \).
• Solve for \( x\): Subtract 4 from both sides to get \( x > -4 \).

Therefore, the domain of the function is \( x > -4 \). This means any x-value greater than \(-4\) will ensure the function is defined and produce a valid output.

Range of a Function

The range of a function refers to all possible outputs a function can produce. For logarithmic functions like \( f(x) = \log_3(x + 4) \), the range is all real numbers. This is because as long as the input values are within the domain, the logarithmic function can generate any real number as a result.

Let’s explore why:

• As \( x + 4 \) approaches just above zero, the output \( \log_3(x + 4) \) becomes very negative.
• As \( x + 4 \) increases, the function value \( \log_3(x + 4) \) also increases without upper bounds.

Hence, as long as you stick to values of \( x > -4 \), \( f(x) \) can be any real number. Because of this property, the range of \( f(x) = \log_3(x + 4) \) is, indeed, all real numbers.

Solving Inequalities

Solving inequalities is a crucial aspect of finding the domain of a function, especially in logarithmic functions. To determine where the function is defined, we set the expression inside the logarithm greater than zero and solve the resulting inequality.

Here's a quick guide:

• Start with the inequality derived from the function's requirement, such as \( x + 4 > 0 \) for \( f(x) = \log_3(x + 4) \).
• Solve for \( x \): by subtracting numbers or making adjustments on both sides without reversing the inequality sign.
• The solution gives the set of all \( x \) values that make the function valid.

When you handle inequalities, remember:

• When multiplying or dividing by a negative number, reverse the inequality sign.
• Focus on isolation of the variable to determine its possible values.

In this particular case, because \( x + 4 > 0 \) simplifies to \( x > -4 \), the inequalities you solve will guide you to determine the domain where the function remains valid and defined.