Problem 32 - A Graphical Approach to Precalculus with Limits

Question

Use the results of the specified exercises to determine (a) the domain and (b) the range of each function. $$y=(x-3)^{2}$$

Step-by-Step Solution

Answer

Domain: \((-\infty, \infty)\), Range: \([0, \infty)\).

Step 1: Identify the Base Function

The function given is \( y = (x - 3)^2 \). This can be recognized as a quadratic function of the form \( y = a(x-h)^2 + k \) where \( a = 1 \), \( h = 3 \), and \( k = 0 \).

Step 2: Determine the Domain

For any quadratic function of the form \( y = (x-h)^2 \), the domain consists of all real numbers because you can plug any real number into the function. Thus, the domain of \( y = (x-3)^2 \) is all real numbers \( x \) or \( (-\infty, \infty) \).

Step 3: Analyze the Range Equation

The range of a function describes the set of possible output values. For \( y = (x-3)^2 \), the expression \((x-3)^2 \) is always non-negative because a square of any real number is non-negative. This means the function values start from \( 0 \) and go to \( \infty \).

Step 4: Define the Range

Since the smallest value \((x-3)^2 \) can take is \(0\) (when \( x = 3 \)), the range of the function \( y = (x-3)^2 \) is \( y \geq 0 \). Thus, the range is \( [0, \infty) \).

Key Concepts

quadratic function

A quadratic function is a type of polynomial function, particularly one where the highest degree is two. In mathematical terms, a quadratic function is typically represented by the standard form:

• \( y = ax^2 + bx + c \)

Here, the coefficient \( a \) cannot be zero, and the other coefficients \( b \) and \( c \) can take any real number values.
The most distinctive characteristic of quadratic functions is their U-shaped graphs, commonly known as parabolas. These parabolas can either open upwards, when \( a \) is positive, or downwards if \( a \) is negative.
When analyzing a quadratic function specifically in vertex form—like our exercise, \( y = (x-h)^2 + k \)—the graph has its highest or lowest point (the vertex) at \( (h, k) \). The function given, \( y = (x-3)^2 \), is in this vertex form, meaning its vertex is at point \( (3, 0) \).
Quadratic functions are quite versatile and appear in a variety of real-world contexts, such as physics, architecture, and economics, particularly when analyzing motion or optimizing certain outcomes.

real numbers

Real numbers encompass a broad category that includes almost every number you can think of. This category spans everything from integers (like -1, 0, 1) to fractions (such as 1/2 or 0.75) and even irrational numbers (like \( \sqrt{2} \) or \( \pi \)).
The notion of real numbers is critical when considering domains of functions. For quadratic functions like \( y = (x-3)^2 \), the domain—the set of possible input values—is all real numbers. This is because you can input any real number value for \( x \), and the function will be able to compute a corresponding value for \( y \). Thus, expressed in interval notation, the domain is \((-\infty, \infty)\).
Understanding real numbers also guides us in comprehending outputs, or ranges, of a function. As in our exercise, we conclude that since squaring any real number yields a non-negative result, the range will start at zero and go up to infinity. Therefore, the range is \([0, \infty)\). Knowing this gives us a backdrop against which we can set realistic expectations for the behavior of other mathematical functions.

function analysis

Function analysis involves dissecting a function to understand its properties and behavior thoroughly. With quadratic functions, this means identifying characteristics such as vertex, symmetry, and range.
For the function \( y = (x-3)^2 \), function analysis starts with identifying the base function as a parabola centered at the vertex \( (3, 0) \). Since \( a = 1 \), the parabola opens upwards, meaning the function only contains non-negative values, establishing the range as \([0, \infty)\).
This form of analysis allows us to study how modifications to the equation, like altering \( h \) or \( k \), impact the graph's positioning and orientation. For instance, changing \( h \) would shift the parabola left or right along the x-axis, while modifying \( k \) would move it up or down along the y-axis.

• Determining the vertex can also help predict outcomes or ideal conditions in real-world applications, like maximizing profit or optimizing resource placement.

Through systematic function analysis, whether it involves parabolas or other graph shapes, we gain the ability to chart inputs and predict outputs with significant accuracy, a cornerstone skill in both mathematical study and practical application.