Problem 20 - College Algebra with Modeling and Visualization

Question

Find the domain of \(f\) and write it in setbuilder or interval notation. $$f(x)=\log (4-\sqrt{2-x})$$

Step-by-Step Solution

Answer

\(x \leq 2\), domain is \((-\infty, 2]\).

Step 1: Understand Logarithm Domain

The domain of a logarithmic function, such as \(\log(a)\), requires that \(a > 0\). This means we need the expression inside the logarithm, \(4 - \sqrt{2-x}\), to be greater than zero.

Step 2: Setup Inequality for Square Root Condition

For the expression \(\sqrt{2-x}\) to be defined, it must be non-negative, i.e., \(2 - x \geq 0\), which implies \(x \leq 2\).

Step 3: Combine Conditions

Combine the conditions from Step 1 and Step 2. From Step 1, \(4 - \sqrt{2-x} > 0\), which simplifies to \(\sqrt{2-x} < 4\). Since a square root is non-negative, this condition is always true as long as the square root is defined. Thus, from Step 2, the condition \(x \leq 2\) still holds.

Step 4: Consider Existence of Square Root

Since \(\sqrt{2-x}\) needs to exist, we confirmed \(x \leq 2\). Therefore, this is the main restriction for the domain of the function \(f(x)\).

Step 5: Write Domain in Setbuilder Notation

The domain, in terms of \(x\), is \(\{ x \mid x \leq 2\} \).

Step 6: Write Domain in Interval Notation

Alternatively, in interval notation, the domain is \((-\infty, 2]\).

Key Concepts

Logarithmic Function

Logarithmic functions are mathematical functions that are the inverse of exponential functions. They are written in the form \(f(x) = \log_b (x)\), where \(b\) is the base of the logarithm. A critical feature of logarithmic functions is that they are only defined for positive real numbers. Therefore, the expression within the logarithm, such as \(4 - \sqrt{2-x}\) in this exercise, must be greater than zero.

• The condition \(a > 0\) ensures that the function will yield real numbers.
• Logarithms return values when the input is positive, as negative inputs or zero do not satisfy the logarithm function's real number domain.

In practical terms, finding the domain of a logarithmic function involves determining where the inside expression is greater than zero, guiding us to the possible values of \(x\) that make the function work.

Inequalities

Inequalities are mathematical expressions that show the relationship of one quantity being greater than or less than another. Solving inequalities helps us find the range of values that satisfy certain conditions.

In the exercise, to keep the logarithmic function defined, the inequality \(4 - \sqrt{2-x} > 0\) is resolved, meaning the term inside must be more than zero. Inequalities have specific rules like linear equations:

• Isolating the variable to find its possible values is the primary goal.
• When multiplying or dividing through an inequality by a negative number, the inequality symbol flips.
• Check conditions like non-negativity for terms such as square roots.

By setting \(x \leq 2\) and combining it with \(\sqrt{2-x} < 4\), the inequality guides us to determine the values of \(x\) that result in a valid expression for \(f(x)\).

Interval Notation

Interval notation is a concise way of representing sets of numbers, commonly used in mathematics to define a range of permissible values. In this form, the domain of a function is compactly described by using brackets and parentheses:

• Square brackets \([ ]\) indicate that the endpoint is included in the interval.
• Parentheses \(( )\) indicate that the endpoint is not included.

For the function \(f(x) = \log (4 - \sqrt{2-x})\), after resolving the inequalities, the domain is expressed in interval notation as \((-\infty, 2]\). This notation indicates that the domain includes all numbers from negative infinity up to and including 2. Being able to read and write interval notation is a powerful tool for expressing domain and range succinctly.

Domain Restrictions

Domain restrictions are conditions that limit the permissible inputs of a function. They are critically important when defining functions because they ensure the function remains valid for all the inputs within its scope.
In the case of a logarithmic function with a square root inside, such as \(f(x) = \log (4 - \sqrt{2-x})\), domain restrictions stem from two main sources:

• The logarithmic function itself, requiring positive inputs.
• The square root function, needing non-negative inputs.

Here, the square root \(\sqrt{2-x}\) brings the restriction \(x \leq 2\), as \(x > 2\) would yield a negative input under the square root, which is undefined in real numbers. Once this is established, it must be cross-verified with the logarithm's positivity condition. Together, these constraints pinpoint the values \(x\) can take, forming the domain limits, and ensuring functional validity.