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Chapter 2: Problem 15

A function \(f\) is given. (a) Use a graphing calculator to draw the graph of \(f .\) (b) Find the domain and range of \(f\) from the graph. $$f(x)=\sqrt{16-x^{2}}$$

Short Answer

Expert verified
Domain: \([-4, 4]\), Range: \([0, 4]\).

Step by step solution

01

Understand the Function

The function is given as \( f(x) = \sqrt{16-x^2} \). This is an equation for a semicircle. The function is defined wherever the expression inside the square root is greater than or equal to zero. Therefore, we need to consider the condition \( 16 - x^2 \geq 0 \).
02

Sketch the Graph Using a Graphing Calculator

Plot the function \( f(x) = \sqrt{16-x^2} \) using a graphing calculator. This should result in an upper semicircle with a radius of 4 centered at the origin on the x-axis, ranging from -4 to 4. The semicircle is above the x-axis.
03

Determine the Domain of \( f \)

The domain of the function consists of x-values for which the expression inside the square root is non-negative. Solving \( 16 - x^2 \geq 0 \), we get \(-4 \leq x \leq 4\). This means the domain of \( f \) is \([-4, 4]\).
04

Determine the Range of \( f \)

The range of \( f \) is determined by the possible y-values that \( f(x) = \sqrt{16-x^2} \) can take. Since this is an upper semicircle of radius 4, the y-values range from 0 to 4. Therefore, the range is \([0, 4]\).
05

Verify with the Graph

Confirm these calculations with the graph obtained from the graphing calculator. The x-values should span from -4 to 4 (domain), and the y-values should span from 0 to 4 (range).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Calculator
A graphing calculator is a powerful tool that helps visualize mathematical functions, including complex calculations. When you're dealing with a function like \( f(x) = \sqrt{16-x^2} \), illustrating it graphically can clarify its characteristics. Luckily, most graphing calculators come with user-friendly interfaces that allow entering equations directly to simulate their graphs.

Here鈥檚 how you can do it:
  • Turn on your graphing calculator.
  • Select the graphing mode, which is often indicated by a 'Graph' or 'Y=' button.
  • Input the function \( \sqrt{16-x^2} \).
  • Adjust the window settings to ensure you capture the entire graph. For this function, you鈥檒l want to set x-axis from -5 to 5 and y-axis from -1 to 5.
  • View the graph. You should see an upper semicircle spanning from x = -4 to x = 4.
Using a graphing calculator not only helps in understanding the range and domain visually but also provides a clearer insight into symmetry and other properties of the function.
Semicircle Function
The function \( f(x) = \sqrt{16-x^2} \) represents a semicircle. A semicircle is half of a circle, which in this case, lies above the x-axis. The equation given is derived from the equation of a circle, \( x^2 + y^2 = 16 \), where the circle has a center at the origin (0, 0) and a radius of 4. By isolating \( y \), where \( y = \sqrt{16-x^2} \), we describe the upper half of the circle.

Important properties of a semicircle function include:
  • Symmetry: The function is symmetric across the y-axis, meaning the left and right halves are mirror images.
  • Domain: The set of x-values for which the function is real. Here, it鈥檚 from \( -4 \) to \( 4 \).
  • Range: The function outputs y-values that vary from \( 0 \) to \( 4 \), representing the height of the semicircle.
This upper semicircle is visually distinct and important in geometry, providing a straightforward visual demonstration of domain and range concepts.
Square Root Function
The square root function, such as \( f(x) = \sqrt{16-x^2} \), undertakes some conditional behavior because the square root requires non-negative numbers. This restriction forms a significant part of determining both domain and range.

Here's what this means:
  • Domain: In \( \sqrt{16-x^2} \), the expression \( 16-x^2 \) must be greater than or equal to zero to ensure that we only take the square root of non-negative numbers. Solving \( 16 - x^2 \geq 0 \) gives the domain \( [-4, 4] \).
  • Range: Since the output of the square root function is non-negative, and the highest value in a semicircle with radius 4 is 4, the range is \( [0, 4] \).
Understanding these limits helps when graphing and analyzing the behavior of functions across different intervals. The square root function is prevalent in various real-world applications due to its origin from geometry and physical phenomena.

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