Problem 104 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function and find its domain and range. $$f(x)=\sqrt{121-x^{2}}$$

Step-by-Step Solution

Answer

The domain of the function \( f(x) = \sqrt{121 - x^{2}} \) is \([ -11, 11 ]\). The range of this function is \([ 0, 11 ]\).

Step 1: Graph the function

Use a graphing utility to graph the function \( f(x) = \sqrt{121 - x^{2}} \). Note that it is the equation of a upper semi-circle with a radius of 11 and centered at the origin.

Step 2: Find the Domain

The domain is all possible x-values. From the graph, you can observe that the function is defined for \( -11 \leq x \leq11 \). So the domain is \([-11,11]\). This is because the square root of a negative number is undefined in the real number system and the equation \( x^{2} \leq 121 \) must be true.

Step 3: Find the Range

The range is all possible y-values. From the graph, you can see that the function produces y-values greater than or equal to 0 and less than or equal to 11. Hence, the range is \([0,11]\). This is because it's the upper half of a circle with a radius of 11.

Key Concepts

Graphing Utilities

Graphing utilities are tools that allow us to visualize mathematical functions and their properties.
They can show how a function behaves and help identify features such as intersecting points, asymptotes, and curves.
Here’s why using a graphing utility can be beneficial:

• Visualization: It turns abstract equations into visual graphics, making it easier to understand the behavior of the function.
• Precision: Graphing utilities can plot equations accurately, helping to identify domain and range.
• Interactivity: Many tools allow for dynamic changes, where you can adjust parameters and instantly see resulting changes in the graph.

In the exercise you have, graphing utilities help us see the function \(f(x) = \sqrt{121 - x^{2}}\) as a half-circle on a coordinate plane. By using these utilities, you can quickly check the correctness of your algebraic solutions to domain and range.

Semi-circle Equation

A semi-circle equation, like the one in your exercise, comes from rearranging the equation of a full circle.
Typically, a circle with a center at the origin is expressed as \(x^2 + y^2 = r^2\), where \(r\) is the radius.
For the semi-circle in the exercise, we have:

• Equation form: \(f(x) = \sqrt{121 - x^{2}}\)
• Circle Radius: \(r = 11\) since \(121 = 11^2\)
• Position: It is centered at the origin \((0,0)\).
• Features: This equation represents the top half (upper semi-circle) of the full circle.

The semi-circle is confined to the upper half-plane due to the square root function, resulting in a range from 0 to the radius, here 11. This helps restrict the y-values to only positive ones in the graph of the semi-circle.

Square Root Function

Square root functions are fundamental in algebra, typically written as \(f(x) = \sqrt{x}\).
This function outputs the principal (non-negative) square root of \(x\).
For your function \(f(x) = \sqrt{121 - x^{2}}\), the square root restricts us to using only values of \(x\) that yield non-negative expressions inside the square root.Important notes about square root functions:

• Domain Limitations: The expression inside the square root must be greater than or equal to zero. Hence, for the exercise, \(121 - x^2 \geq 0\) is needed which means \(-11 \leq x \leq 11\).
• Range Constraints: The square root ensures that the smallest output is zero. In the context of a semi-circle, the largest value is the radius, here being 11.
• Graph Characteristics: The graph is curved, sloping upwards and never going below the x-axis, reflecting only positive y-values.

By understanding how square root functions work, you can interpret and analyze real-world scenarios where only non-negative outcomes are reasonable.