Problem 1 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Specify the domain of the function $$ f(x)=\frac{2 x-1}{\sqrt{4-x^{2}}} $$

Step-by-Step Solution

Answer

The domain of the function is (-2, 2).

Step 1: Identify conditions for the denominator

The function has a fraction with a denominator containing a square root. For the square root to be a real number, the expression inside it must be non-negative (since the square root of a negative number is undefined in the set of real numbers). So, we need to solve the inequality: The square root function is defined when the expression under the radical is positive: 4 - x^2 > 0

Step 2: Solve the inequality

Solve the inequality found in Step 1. First, rewrite the inequality: 4 - x^2 > 02. Isolate the term involving x: -x^2 > -4.2. Then, divide both sides by -1 (remember to reverse the inequality sign when dividing by a negative number): x^2 < 4.Taking the square root of both sides results in: -2 < x < 2.

Step 3: Exclude points where the denominator becomes zero

The function is undefined where the denominator is zero. Thus, for The function changes its slope when the denominator is zero: sqrt (4 - x^2) = 0,4 - x^2 = 0,x^2 = 4,Solving this gives the points x = -2 and x =2.Therefore, these points must be excluded from the domain.

Step 4: Combine the results

Taking all conditions into account, the domain of f(x) excludes the points x = -2 and x =2 but includes all values between them. Thus, the domain of f(x) is: xi n( -2, 2).

Key Concepts

Inequalities in Calculus

When working with calculus and functions, inequalities help us determine the range of valid input values (domain) for which a function is defined. In the provided exercise, we encounter the inequality involving the term under the square root: 4 - x^2 > 0This inequality is critical because the square root function is only defined for non-negative numbers. Here are the steps to solve this inequality:

• Rearrange the inequality: -x² > -4
• Divide by -1 (remember to flip the inequality sign): x² < 4
• Take the square root of both sides: -2 < x < 2

The solution to this inequality helps us define the preliminary range of x-values for which the function is real and valid, before taking any further restrictions into account.

Fractional Functions

Fractional functions are functions that contain fractions where the numerator and/or the denominator involve variables. In our example, the function is: \[ f(x) = \frac{2x - 1}{\text{sqrt}(4 - x^2)} \] For fractional functions, it is essential that the denominator does not become zero, because division by zero is undefined. In this case, the denominator is \( \text{sqrt}(4 - x^2) \). From earlier, we know that the denominator is defined if 4 - x^2 > 0. However, we also need to ensure 4 - x^2 ≠ 0 because that would make the denominator zero. Solving 4 - x^2 = 0 gives us x = ±2, which we must exclude from our domain, narrowing it down further.

Domain Restrictions

Domain restrictions are constraints on the input values of a function to ensure it is well-defined (non-zero denominators and real numbers under square roots, etc.). For the function \( f(x) = \frac{2x - 1}{\text{sqrt}(4 - x^2)} \), we face multiple restrictions:

• The inequality we solved previously: -2 < x < 2
• Excluding points where the denominator zeroes out: x ≠ -2 and x ≠ 2

Combining these, we see that our restrictions both set the range and exclude specific points within that range: -2 < x < 2, but without x = -2 and x = 2, which leaves us with the domain: \( x \text{∈} (-2, 2) \).