Problem 79 - Precalculus

Question

Determine the domain and range of the quadratic function. \(f(x)=(x-3)^{2}+2\)

Step-by-Step Solution

Answer

Domain: \( (-\infty, +\infty) \), Range: \( [2, +\infty) \).

Step 1: Understand the Quadratic Function

We have the function given by \( f(x) = (x-3)^2 + 2 \). This is a quadratic function in vertex form, where \( (x-3)^2 \) indicates it's a parabola that opens upwards. The term \( (x-3) \) indicates a horizontal shift to the right by 3 units, and the \( +2 \) indicates a vertical shift upwards by 2 units.

Step 2: Determine the Domain

The domain of a function includes all possible input values, \(x\). For any quadratic function, \(f(x)=(x-3)^2+2\), \(x\) can be any real number. Therefore, the domain is all real numbers, denoted by \( \text{Domain}: \mathbb{R} \) or \( (-\infty, +\infty) \).

Step 3: Determine the Range

The range consists of all possible output values, \(f(x)\). Since \(f(x) = (x-3)^2 + 2\) opens upwards, the minimum value occurs at the vertex. The vertex is at \(x=3\) and \(f(3) = (3-3)^2 + 2 = 2\). Thus, the range is \(f(x) \geq 2\). Therefore, the range is \( \text{Range}: [2, +\infty) \).

Key Concepts

Domain of a Quadratic Function

Understanding the concept of the "domain" is crucial in mathematics. When we speak of the domain, we are referring to all the possible input values (in this case, of a function). For a quadratic function, which generally takes the form \( f(x) = ax^2 + bx + c \), the domain is concerned with the values that \( x \) can take.

For a standard quadratic function like our original exercise, \( f(x)=(x-3)^2+2 \), we are looking at any real number that can be plugged into \( x \). This is because the expression \( (x-3)^2 \) will yield a result for any value of \( x \). Thus:

• The domain encompasses all real numbers.
• This is written formally as \( \mathbb{R} \) or \( (-\infty, +\infty) \).

This understanding highlights an inherent property of quadratic functions—they are defined for every real number.

Range of a Quadratic Function

Similar to the domain, the "range" describes the set of all possible output values. For the quadratic function given, \( f(x) = (x-3)^2 + 2 \), the range is determined by the behavior of the parabola, particularly focusing on the vertex.

The vertex of this quadratic function occurs at the point \( x=3 \). Plugging this back into the function gives \( f(3) = (3-3)^2 + 2 = 2 \). When the parabola opens upwards, which it does in this function, the vertex represents the minimum point. Thus, every value \( f(x) \) takes is greater than or equal to this minimum.

• This means the range starts at 2 and extends to infinity.
• This is expressed as \( [2, +\infty) \). It includes 2 and all values greater than 2.

Understanding the range helps you know what output values are possible for every input value, clarifying the complete picture of the function's behavior.

Vertex Form of a Quadratic Function

The "vertex form" of a quadratic function provides insights into the parabola's orientation and position. The formula \( f(x) = a(x-h)^2 + k \) captures this, where \( (h, k) \) is the vertex of the parabola.

In the example \( f(x) = (x-3)^2 + 2 \), you can clearly see this format with \( h=3 \) and \( k=2 \). These values show us:

• The parabola's vertex is at the point (3,2).
• The parabola is shifted right by 3 units and up by 2 units from the origin.

Additionally, because the coefficient of \( (x-3)^2 \) (which is 1) is positive, the parabola opens upwards. This vertex form is particularly handy because it makes identifying the vertex and understanding transformations straightforward. By mastering this form, you can easily assess the nature and location of the graph of a quadratic function.