Problem 13 - A Graphical Approach to College Algebra

Question

Find the domain of each logarithmic function analytically. You may wish to support your answer graphically. $$y=\ln \left(x^{2}+7\right)$$

Step-by-Step Solution

Answer

The domain of \( y = \ln(x^2 + 7) \) is all real numbers.

Step 1: Understanding the Logarithmic Function

To find the domain of a logarithmic function like \( y = \ln(x^2 + 7) \), remember that the argument of the logarithm (\( x^2 + 7 \)) must be greater than zero. The natural logarithm is undefined for non-positive numbers.

Step 2: Setting Up the Inequality

Since the argument \( x^2 + 7 \) must be greater than zero for the logarithm to be defined, set up the inequality: \( x^2 + 7 > 0 \).

Step 3: Solving the Inequality

Solve \( x^2 + 7 > 0 \):- The expression \( x^2 + 7 \) is always greater than zero for any real \( x \) because \( x^2 \) is non-negative and adding 7 ensures it is always positive.- Therefore, the inequality is true for all real numbers \( x \).

Step 4: Determining the Domain

Since \( x^2 + 7 > 0 \) for all real \( x \), the domain of the function \( y = \ln(x^2 + 7) \) is all real numbers.

Key Concepts

Domain of a Function

The domain of a function refers to the set of all possible input values (usually denoted as \( x \)) that allow the function to work properly. When dealing with a logarithmic function like \( y = \ln(x^2 + 7) \), it is crucial to understand what values \( x \) can hold to keep the function \( y \) defined and without errors.
The natural logarithm, \( \ln \), functions only when its argument is positive. In this case, it means \( x^2 + 7 \) must be greater than zero. Let's break this down:

• The expression \( x^2 \) is never negative — it is zero or positive regardless of the value of \( x \).
• By adding 7 to \( x^2 \), we ensure the expression \( x^2 + 7 \) is always positive.

Therefore, for the function \( y = \ln(x^2 + 7) \), every real number \( x \) is a valid input. Hence, the domain is all real numbers. This means you can plug any real number into \( x \) and get a valid output for \( y \).

Inequalities

Inequalities are mathematical expressions that show the relationship between two quantities that are not equal. In this exercise, inequalities help us find which values of \( x \) make \( x^2 + 7 \) a positive number, allowing the logarithm to exist.
Consider the inequality \( x^2 + 7 > 0 \):

• This inequality states that whatever value \( x \) holds, when squared and added to 7, the result must be greater than zero.
• Since squaring any real number results in zero or a positive value, \( x^2 + 7 \) inherently remains positive.
• Therefore, no specific restrictions on \( x \) need to be set, as the inequality holds true for all real numbers.

Understanding how inequalities work helps us define the domain of functions, particularly when ensuring the logarithm's argument is positive.

Real Numbers

Real numbers encompass all the numbers you might typically consider for daily tasks and advanced mathematics. This includes positive numbers, negative numbers, fractions, and irrational numbers like \( \pi \) and \( \sqrt{2} \).
Real numbers are important to the concept of the domain in functions because they help define the possible inputs. For our function \( y = \ln(x^2 + 7) \):

• Since \( x^2 + 7 \) remains positive regardless of \( x \)'s value, any real number can be used for \( x \).
• This implies there are no breaks, gaps, or holes in the set of numbers \( x \) can take to keep the function defined.

Thus, when considering the domain of \( y = \ln(x^2 + 7) \), we recognize that it is based on the characteristic of real numbers, allowing \( x \) to take any real value, emphasizing how robust and unbounded real numbers are when applied correctly in mathematical functions.