Problem 106 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function and find its domain and range. $$f(x)=-x^{2}+9$$

Step-by-Step Solution

Answer

The domain of the function is \(x \in (-\infty, \infty)\). The range of the function is \(y \in (-\infty, 9]\).

Step 1: Graph the Function

Start by graphing the function \(f(x)=-x^{2}+9\). Remember that this is a standard quadratic function, which means the graph will be a parabola. However, because the coefficient of \(x^{2}\) is negative, the parabola will be inverted, or 'upside down'.

Step 2: Determine the Domain

The domain of a function is the set of all possible x-values. For polynomial functions, the domain is always all real numbers. That is because you can input any real number into the function and get a real number output. Therefore, the domain of this function is \(x \in (-\infty, \infty)\)

Step 3: Determine the Range

The range of a function is the set of all possible y-values. Because this is an inverted parabola, the range will be all y-values that are less than or equal to the maximum value of the function. The maximum value occurs at the vertex of the parabola. In this case, the vertex is at y = 9. Therefore, the range of the function is \(y \in (-\infty, 9]\)

Key Concepts

Understanding the Parabola

A parabola is the U-shaped graph that represents a quadratic function. In the given exercise, the quadratic function is presented in standard form as \(f(x) = -x^2 + 9\). This means we are dealing with a polynomial function of degree 2, which invariably graphs as a parabola.
There are some essential properties of parabolas:

• Vertex: The point at which the parabola changes direction. For our equation, the vertex is at the origin if it were not shifted up or down. However, due to the \(+9\), it is at (0, 9).


• Axis of Symmetry: A vertical line that divides the parabola into two mirror-image halves. This is the x-value of the vertex, which here is \(x = 0\).


• Direction: Since the leading coefficient (in front of \(x^2\)) is negative, our parabola opens downwards. If it were positive, it would open upwards.

In our example, this inverted parabola signifies that the vertex at (0, 9) is the maximum point.

Exploring Domain and Range

The domain and range are critical concepts when graphing functions. Let's break these down for our function \(f(x) = -x^2 + 9\).
Domain: The domain refers to all possible x-values that you can input into the function. Polynomial functions, like our quadratic one, have a domain of all real numbers, \(x \in (-\infty, \infty)\). This means you can substitute any real number for \(x\) and get a valid output.
Range: The range is the set of possible y-values. For our inverted parabola, the vertex represents the highest point on the graph. Thus, all other y-values we obtain will be less than or equal to 9, the y-coordinate of the vertex. Consequently, our range is \(y \in (-\infty, 9]\).
These concepts help in fully describing the behavior of the quadratic function over all possible values.

Polynomial Functions and Their Properties

Polynomial functions are diverse and exhibit various degrees and shapes. They include terms that are powers of \(x\), and their general form is \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\).
For our specific polynomial \(f(x) = -x^2 + 9\):

• It's a quadratic function, meaning it has a degree of 2, indicating the highest power of \(x\) in the equation is 2.


• The leading coefficient here is \(-1\). This impacts the function in several ways, primarily that the parabola opens downwards.


• Polynomials are continuous functions. They don't have breaks or gaps in their graph, and this is why their domain includes all real numbers.

Understanding these properties helps in analyzing the behavior of polynomial functions and how they can be graphed.