Problem 36 - Intermediate Algebra

Question

Specify the domain for each of the functions. $$f(x)=\sqrt{9-x^{2}}$$

Step-by-Step Solution

Answer

The domain is \([-3, 3]\).

Step 1: Understand the Function

The given function is \( f(x) = \sqrt{9-x^2} \). It is a square root function. The expression inside the square root must be non-negative for the function to be defined.

Step 2: Set Up the Inequality

To ensure the expression \( 9-x^2 \) is non-negative, we set up the inequality \( 9-x^2 \geq 0 \). Solving this inequality will give us the domain of the function.

Step 3: Solve the Inequality

Rearrange the inequality: \( 9 \geq x^2 \). This implies that \( -3 \leq x \leq 3 \), since taking the square root of both sides gives the range of \( x \) that satisfies this inequality.

Step 4: State the Domain

Thus, the domain of \( f(x) = \sqrt{9-x^2} \) is all the \( x \) values for which the inequality is true. Therefore, the domain is \([-3, 3]\).

Key Concepts

Domain of a function

The domain of a function is the complete set of values that the input, typically represented by \( x \), can take for which the function is defined. Understanding the domain is crucial because it tells us where the function exists and can operate without issues such as division by zero or taking square roots of negative numbers.
When determining the domain, think about any constraints that the function might have. These constraints come from mathematical rules like you cannot divide by zero or take the square root of a negative number. The domain is essentially where every part of the function makes mathematical sense.

• Identify any mathematical operations that might limit input values. For example, square roots require non-negative numbers.
• Consider if there's division in the function, as division by zero is undefined.

In our case, we are focusing on the square root function, which specifically requires that the value under the root is zero or positive for real number outputs.
For the function \( f(x) = \sqrt{9-x^2} \), the domain will be found by setting up an inequality problem that ensures the expression inside the square root is greater than or equal to zero.

Square root function

The square root function is a special type of function where a number \( x \) is multiplied by itself to form another number. In function form, the square root is written as \( f(x) = \sqrt{x} \).
In the context of our problem, the square root function is \( f(x) = \sqrt{9-x^2} \). For the square root to be defined and real, the expression inside \( \sqrt{} \) must not be negative. This results in an inequality which helps us find the valid range of \( x \).
It's essential to realize that while square roots can handle zero and positive numbers, they cannot naturally take negative inputs and still produce real number outcomes. Problems involving square roots often revolve around ensuring the square root expression's "argument" (what's inside the root) is non-negative.
Let's see this reasoning applied to our initial function. Given \( 9-x^2 \), this expression must be zero or positive.

• If \( 9-x^2 = 0 \), then \( x \) is 3 or -3. Both are part of the domain.
• If \( 9-x^2 > 0 \), then values of \( x \) between -3 and 3 also work.

Hence, square root functions like these often become exercises in expressing interval notation of domain restrictions.

Inequalities in algebra

Inequalities are used in algebra to express a range of values that are possible for a variable. It's an extension of equality (\( = \)) by permitting less than (\( < \)), greater than (\( > \)), less than or equal to (\( \leq \)), and greater than or equal to (\( \geq \)) relationships.
To solve inequalities, you follow similar steps to solving regular equations, like combining like terms and isolating the variable on one side. However, one must pay special attention when multiplying or dividing by negative numbers, as this will flip the inequality sign.
For solving \( 9-x^2 \geq 0 \), we rearrange to \( 9 \geq x^2 \). Solving for \( x \), we determine the points where equality holds \( x^2 = 9 \), which are \( x = 3 \) and \( x = -3 \).
Since \( x^2 \) represents a parabola opening upwards, and \( 9 \geq x^2 \) defines the interval where the function touches or stays below 9, the solution is the set of \( x \) bounded by \( -3 \leq x \leq 3 \).

• Replace the inequality with equality to find critical points.
• Tests points around the critical values to determine which intervals conform to the original inequality.

This ensures understanding of how these mathematical expressions dictate the region where the function lives, providing a complete picture of the solution's allowable range in function terminology.