Problem 40 - Calculus with Concepts in Calculus

Question

Determine the range of the function. $$ f(x)=\sqrt{1-x^{2}} $$

Step-by-Step Solution

Answer

The range of the function is \([0, 1]\).

Step 1: Understand the Domain of the Function

The function \( f(x) = \sqrt{1-x^2} \) contains a square root, which is only defined for non-negative values. Thus, the expression under the square root, \( 1-x^2 \), must be non-negative. This gives us the inequality \( 1-x^2 \geq 0 \). Solving this inequality helps us find the domain of the function, which indirectly helps in determining the range.

Step 2: Solve the Inequality

Solve the inequality \( 1-x^2 \geq 0 \). Rearrange to \( x^2 \leq 1 \). This means \( x \) is between -1 and 1, inclusive. Therefore, the domain of \( f(x) \) is \( -1 \leq x \leq 1 \).

Step 3: Analyze the Output of the Function

For the values in the domain \( -1 \leq x \leq 1 \), calculate the range of the function. Substitute the boundary values into \( f(x) \) to identify the smallest and largest possible values of \( f(x) \).

Step 4: Calculate Function Values at Boundary Points

Calculate \( f(-1) = \sqrt{1-(-1)^2} = \sqrt{0} = 0 \) and \( f(1) = \sqrt{1-(1)^2} = \sqrt{0} = 0 \). Since \( f(x) = \sqrt{1-x^2} \), the function reaches its maximum value when \( x = 0 \). Calculate \( f(0) = \sqrt{1-(0)^2} = \sqrt{1} = 1 \).

Step 5: Determine the Range

Since \( 0 \leq \sqrt{1-x^2} \leq 1 \) for \( -1 \leq x \leq 1 \), the range of \( f(x) \) is from 0 to 1, inclusive.

Key Concepts

Domain Determination

The domain of a function comprises all the values that can be inputted into the function, leading to valid outputs. When dealing with functions that include square roots, like \( f(x) = \sqrt{1-x^2} \), it is crucial to understand that the expression inside the square root must be non-negative. This is because square roots of negative numbers aren't defined in the realm of real numbers. To determine the domain, we set the expression inside the square root, \( 1 - x^2 \), to be greater than or equal to zero. This step ensures that the function will yield real number outputs for any input \( x \).

Thus, we solve the inequality \( 1 - x^2 \geq 0 \). Rearranging, we get \( x^2 \leq 1 \), meaning \( x \) can take any value between -1 and 1, inclusive. Hence, the valid domain of the function is the closed interval \([-1, 1]\). This insight is instrumental in finding the function's range.

Inequality Solving

Solving inequalities is a fundamental skill for determining domains, especially with functions involving roots or fractions. Here, for \( f(x) = \sqrt{1-x^2} \), after establishing the inequality \( 1-x^2 \geq 0 \), we aimed to find the allowed values of \( x \).

To tackle \( 1 - x^2 \geq 0 \), we reorganize it to \( x^2 \leq 1 \). Solving these types of inequalities involves a few straightforward steps.

• First, recognize that \( x^2 \leq 1 \) implies \( x \leq 1 \) and \( x \geq -1 \).
• This characterizes the interval \( -1 \leq x \leq 1 \), meaning \( x \) can be any value within this range.
• Thus, the inequality solution gives us valid \( x \) values for which the function is defined.

This step ensures that you precisely understand the scope of input values, facilitating deeper analysis in subsequent steps.

Boundary Analysis

Analyzing boundaries is an essential step in identifying a function's range. Boundaries for our function \( f(x) = \sqrt{1-x^2} \) come from the domain, which we established as \([-1, 1]\).

To assess behavior at the boundary values, calculate the function values at the endpoints of the domain. Start by substituting \( x = -1 \) and \( x = 1 \) into the function:

• \( f(-1) = \sqrt{1 - (-1)^2} = \sqrt{0} = 0 \)
• \( f(1) = \sqrt{1 - 1^2} = \sqrt{0} = 0 \)

Additionally, check the function's behavior at the midpoint \( x = 0 \):

• \( f(0) = \sqrt{1 - 0^2} = \sqrt{1} = 1 \)

These calculations show the behavior at and within boundary points, verifying the extremities of the function range. Boundary analysis confirms that the smallest output is 0 and the largest is 1.

Function Evaluation

Function evaluation involves determining the actual outputs of a function for a given domain. For \( f(x) = \sqrt{1-x^2} \), once the domain \([-1, 1]\) was established, it becomes straightforward to verify and understand the range.

By substituting the extreme boundary values \( -1 \) and \( 1 \), and midpoint \( 0 \) into the function, you observe:

• Both \( f(-1) \) and \( f(1) \) result in 0, as the square root of zero is zero.
• At the midpoint \( x = 0 \), the output is 1, showcasing the function's maximum.

This analysis confirms our conclusions from boundary analysis. Hence, the range of \( f(x) = \sqrt{1-x^2} \) is \([0, 1]\), encompassing all potential function outputs from the minimum to the maximum within the specified domain.