Problem 42 - Algebra and Trigonometry

Question

\(41-44=\) Find the domain and range of the function. $$ f(x)=x^{2}-2 x-3 $$

Step-by-Step Solution

Answer

Domain: (-∞, ∞); Range: [-4, ∞).

Step 1: Understand the function

The given function is a quadratic function, which is in the standard form \( f(x) = ax^2 + bx + c \). Here, \( a = 1 \), \( b = -2 \), and \( c = -3 \). Quadratic functions have a parabolic shape.

Step 2: Find the domain

The domain of a function is the set of all possible input values (\( x \) values). For quadratic functions, which are polynomials, the domain is all real numbers, since there are no restrictions on the \( x \) values. Thus, the domain is \( (-\infty, \infty) \).

Step 3: Determine the vertex

The vertex of a quadratic function given in standard form \( ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \). Here, \( a = 1 \) and \( b = -2 \), so the x-coordinate of the vertex is \( x = -\frac{-2}{2 \times 1} = 1 \).

Step 4: Find the range

The range is determined by the vertex of the parabola. Substitute \( x = 1 \) into the function to find the y-coordinate of the vertex: \( f(1) = (1)^2 - 2(1) - 3 = 1 - 2 - 3 = -4 \). Since the parabola opens upwards (\( a > 0 \)), the range of \( f(x) \) is all values greater than or equal to -4. Thus, the range is \( [-4, \infty) \).

Key Concepts

Quadratic Functions

The function given in the exercise, \( f(x) = x^2 - 2x - 3 \), is an example of a quadratic function. Quadratic functions are represented by the general form:

• \( f(x) = ax^2 + bx + c \)

where \( a \), \( b \), and \( c \) are constants. In our function, these constants are:

• \( a = 1 \)
• \( b = -2 \)
• \( c = -3 \)

Quadratic functions are interesting because they describe a symmetrical curve that can be graphed as a U-shape called a parabola. The coefficient \( a \) plays a crucial role, as it determines the direction and width of the parabola. If \( a > 0 \) like in our example, the parabola opens upwards. Otherwise, it would open downwards if \( a < 0 \). Understanding these features helps in predicting the behavior of the curve just by analyzing the function.

Parabolic Shape

The graph of a quadratic function such as \( f(x) = x^2 - 2x - 3 \) forms a curve known as a parabola. This specific curve has a symmetric shape:

• It is mirror-symmetrical around a vertical line called the axis of symmetry.

Parabolas are crucial in helping us understand and visualize quadratic equations. Part of their beauty lies in their simplicity:

• The vertex is the highest or lowest point of the parabola.
• The y-intercept is easy to find by setting \( x = 0 \).
• The x-intercepts can be found through factoring or using the quadratic formula.

In this problem, our parabola opens upward since \( a = 1 \) is positive. This means that the vertex will be the lowest point in this curve. Knowing the parabola's structure helps with finding the domain and range and analyzing the changes in the function's values.

Vertex Formula

A critical part of understanding quadratics like \( f(x) = x^2 - 2x - 3 \) is finding the vertex. We use the vertex formula:

• \( x = -\frac{b}{2a} \)

This helps locate the x-coordinate of the vertex efficiently. Substituting \( a = 1 \) and \( b = -2 \) from our equation, we find:

• \( x = -\frac{-2}{2 \times 1} = 1 \)

The vertex \( x \)-coordinate allows us to further find the \( y \)-coordinate by plugging back into the original equation:

• \( f(1) = 1^2 - 2(1) - 3 = -4 \)

Hence, the vertex of the parabola is \( (1, -4) \). This is a pivotal point on the parabola, assisting in defining the range and indicating the minimum or maximum value of the function.

Real Numbers

When discussing quadratic functions and their domain, understanding real numbers is fundamental. The domain of a polynomial function like \( f(x) = x^2 - 2x - 3 \), specifically a quadratic, is composed of all real numbers. This means:

• Every real number \( x \) can be an input.

Real numbers encompass:

• Integers (positive, negative, and zero)
• Fractions
• Decimals
• Irrational numbers

They constitute an unbroken continuum along the number line, resulting in quadratic functions having continuous graphs without any breaks or holes. Because of this continuity, defining the domain as all real numbers \((-\infty, \infty)\) is a natural conclusion for quadratic functions.