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Chapter 3: Problem 16

\(9-18\) m 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. $$ f(x)=-\sqrt{25-x^{2}} $$

Short Answer

Expert verified
Domain: \([-5, 5]\); Range: \([-5, 0]\).

Step by step solution

01

Understand the Function

The function given is \( f(x) = -\sqrt{25-x^2} \). This means for each \( x \), the function outputs the negative square root of \( 25-x^2 \). This is a semi-circle function in the negative y-direction.
02

Determine the Domain Algebraically

Since \( f(x) = -\sqrt{25-x^2} \), the expression inside the square root must be non-negative: \( 25-x^2 \geq 0 \). Solving \( -5 \leq x \leq 5 \), we get the domain of \( f(x) \).
03

Graph the Function

Using a graphing calculator, graph \( f(x) = -\sqrt{25-x^2} \). It will look like a semicircle facing downwards from \( x = -5 \) to \( x = 5 \).
04

Find the Domain from the Graph

From the graph, verify the domain is from \(-5\) to \(5\) along the x-axis. This corresponds to all x-values from -5 to 5 inclusive where the function is defined.
05

Determine the Range from the Graph

Observe the y-values on the graph. Since this is a negative square root function, the highest value it reaches is 0 and it decreases to -5 at its lowest (bottom of the semicircle). Thus, the range is \(-5 \leq y \leq 0\).

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Key Concepts

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

Understanding Domain and Range
The domain and range of a function are crucial concepts in mathematics.
This understanding helps to identify the set of possible input and output values a function can take.

The **domain** refers to all the possible x-values that can be plugged into a function without causing any inconsistencies, such as division by zero or taking the square root of a negative number.
  • For the function \(f(x) = -\sqrt{25-x^2}\), we explore when \(25-x^2\) is non-negative because we cannot square root a negative number.
  • As a result, the inequality \(25-x^2 \geq 0\) leads to the domain \(-5 \leq x \leq 5\).

The **range**, however, concerns all possible y-values that the function can output.
  • For \(f(x) = -\sqrt{25-x^2}\), since we have a negative square root, the maximum value is 0 (when \(x = \pm 5\) and \(25-x^2=0\)), and it decreases down to -5 (when \(x = 0\), and \(25-x^2=25\)).
  • This means the range is \(-5 \leq y \leq 0\).
Exploring Square Root Functions
Square root functions are a type of radical function characterized by the presence of a square root.
These functions are key in representing many natural and mathematical phenomena.

The general form of a square root function is \(f(x) = \sqrt{g(x)}\), where \(g(x)\) must be non-negative to ensure real outputs (as we cannot consider square roots of negative numbers in real numbers).
  • In the expression \(f(x) = -\sqrt{25-x^2}\), we encounter a square root as part of the function.
  • The negative sign in front of the square root flips the output values in the vertical direction, affecting the range.

Square root functions often portray a graph that represents half of a parabola.
  • In this specific function, the transformation happens in the vertical direction, resulting in a downward U-shape, indicating half of a *circle* rather than a parabola.
Visualizing the Semicircle Graph
When graphed, \(f(x) = -\sqrt{25-x^2}\) represents a semicircle.
However, unlike a typical circle, this semicircle opens downwards.

This semicircle's graph provides a visual curve that spans just one half of a complete circle, determined by the domain.
  • The center of this semicircle is at the origin, but since it is negative, the entire curve is below the x-axis from \(x = -5\) to \(x = 5\).
  • The endpoints, where the curve meets the x-axis, are at the points \((-5,0)\) and \((5,0)\), showing that these are boundary conditions for the function.

To fully understand this input-output relationship visually, it's beneficial to use a graphing calculator.
  • With graphing tools, you can better visualize how the semicircle descends from the origin to a minimum y-value of -5 when \(x = 0\).
  • Seeing the graph can solidify your understanding of its properties including its domain of \(-5 \leq x \leq 5\) and range of \(-5 \leq y \leq 0\).

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