Problem 32 - A Graphical Approach to College Algebra

Question

Use the results of the specified exercises to determine (a) the domain and (b) the range of each function. \(\left.y=(x-3)^{2} \text { (Exercise } 18\right)\)

Step-by-Step Solution

Answer

The domain is all real numbers; the range is \( y \geq 0 \).

Step 1: Understand the Function

The function given is \( y = (x-3)^2 \). This is a quadratic function, which is a type of polynomial function that forms a parabola when graphed. The parabola opens upwards because the coefficient of the \( x^2 \) term is positive.

Step 2: Determine the Domain

The domain of a function is the set of all possible input values (\( x \)-values) for which the function is defined. For the function \( y = (x-3)^2 \), there are no restrictions on the values of \( x \) because polynomials are defined for all real numbers. Therefore, the domain is all real numbers, or \( x \in (-\infty, \infty) \).

Step 3: Analyze the Range

The range of a function is the set of all possible output values (\( y \)-values). For \( y = (x-3)^2 \), the smallest value of \( y \) occurs when \( (x-3)^2 = 0 \). Solving \( (x-3)^2 = 0 \) gives \( x = 3 \), yielding \( y = 0 \). Since \( (x-3)^2 \) is always non-negative, \( y \) cannot be negative. Thus, the range of the function is all non-negative real numbers, or \( y \in [0, \infty) \).

Key Concepts

Domain of a Function

The domain of a function is essentially the collection of all possible input values (commonly represented as the variable \( x \)) for which the function is mathematically defined. In simpler terms, it's the complete set of all \( x \)-values you can plug into the function without encountering mathematical hitches like division by zero or taking a square root of a negative number.

For a polynomial function like a quadratic, which our original exercise deals with, determining the domain is straightforward because polynomial functions are defined for all real numbers. This makes them very predictable and easy to work with. Specifically, quadratic functions like \( y = (x-3)^2 \) can accept any \( x \)-value.

• There are no restrictions like there would be for a square root or a fraction.
• The domain is represented as \( (-\infty, \infty) \), suggesting that you can choose any real number for \( x \), from negative infinity to positive infinity.

Understanding the domain is crucial as it tells you the complete range of inputs your function can handle. Knowing this helps you avoid errors in calculations or unexpected results when analyzing or graphing the function.

Range of a Function

The range of a function gives us valuable insight into its behavior by showing all possible output values (commonly represented by \( y \)). It's where we see the values that the function can achieve once all potential inputs (\( x \)-values) in the domain have been evaluated.

In the case of quadratic functions like \( y = (x-3)^2 \), the outputs are always non-negative. This is because squaring any real number (positive or negative) results in a non-negative product. Let's break it down:

• At \( x = 3 \), the expression \( (x-3)^2 = 0 \), so \( y = 0 \). This is the lowest value achievable by the function.
• For any other value of \( x \), \( y \) becomes larger than zero, starting from zero and moving upwards indefinitely.

The range for this function is all non-negative real numbers, denoted as \( [0, \infty) \). This means the smallest value \( y \) can have is zero, but there is no upper limit to the values \( y \) can take. Grasping the range is key to understanding how the outputs of the function behave and how they can be graphically represented.

Polynomial Functions

Polynomial functions are a crucial part of algebra as they represent a broad class of equations involving powers of \( x \). They are typically expressed in the form \( P(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0 \), where each coefficient \( a_i \) is a constant and the powers of \( x \) are whole numbers.

Quadratic functions, like \( y = (x-3)^2 \), are particularly prevalent as they are polynomials of degree 2. Such functions have a parabolic graph which can open upwards or downwards depending on the sign of the leading coefficient. Here, since it's positive, the graph forms an upward opening parabola.

• Polynomial functions are continuous, meaning there are no breaks or holes in their graphs.
• They are smooth curves, which makes them nice to work with compared to functions with sharp corners or discontinuations.
• Because of their predictive nature across all real \( x \)-values, they often serve in modeling physical phenomena.

Understanding polynomial functions like quadratics will arm you with tools to tackle a variety of real-world and theoretical problems efficiently, reinforcing the way you interpret graphs and solve algebraic equations.