Problem 47 - Algebra and Trigonometry

Question

Find the domain and range of the function. $$ f(x)=2 x^{2}+6 x-7 $$

Step-by-Step Solution

Answer

The domain is \( (-\infty, \infty) \) and the range is \( \left[ -\frac{7}{2}, \infty \right) \).

Step 1: Understanding the Function Type

The function given is a quadratic function of the form \( f(x) = ax^2 + bx + c \) where \( a = 2 \), \( b = 6 \), and \( c = -7 \). Quadratic functions generally have a parabolic shape and open upwards if \( a > 0 \).

Step 2: Finding the Domain of the Function

The domain of a quadratic function is all real numbers because you can substitute any real number for \( x \) without restriction. Thus, the domain of \( f(x) = 2x^2 + 6x - 7 \) is all real numbers, expressed as \( (-\infty, \infty) \).

Step 3: Determining the Vertex

To find the range, identify the vertex of the parabola. The vertex formula for \( x \ ext{-coordinate} \) of a quadratic function is \( x = -\frac{b}{2a} \). Substituting \( b = 6 \) and \( a = 2 \), we get \( x = -\frac{6}{4} = -\frac{3}{2} \).

Step 4: Computing the Vertex Value

Substitute \( x = -\frac{3}{2} \) into the function to find the \( y \ ext{-coordinate} \). Compute \( f\left(-\frac{3}{2}\right) = 2\left( -\frac{3}{2} \right)^2 + 6\left( -\frac{3}{2} \right) - 7 \). This simplifies to \( 2 \times \frac{9}{4} - 9 - 7 = \frac{9}{2} - 16 = \frac{-7}{2} \). So the vertex is \( \left(-\frac{3}{2}, -\frac{7}{2}\right) \).

Step 5: Finding the Range of the Function

Since the parabola opens upwards (\( a > 0 \)), the range is all values greater than or equal to the \( y \ ext{-coordinate} \) of the vertex, which is \( -\frac{7}{2} \). Therefore, the range is \( \left[ -\frac{7}{2}, \infty \right) \).

Key Concepts

Domain and Range

When dealing with quadratic functions, one of the key concepts to understand is the domain and range. The **domain** of a quadratic function refers to all the possible values that can be input into the function. For any quadratic function, you can substitute any real number for \( x \) without any restrictions. Therefore, the domain of any quadratic function is all real numbers, denoted as \((-\infty, \infty)\).

The **range** of a quadratic function, on the other hand, refers to all the possible values that the function can output. Since the quadratic function is a parabola, the direction in which it opens (upwards or downwards) will influence the range. For a function like \( f(x) = 2x^2 + 6x - 7 \), where \( a > 0 \), the parabola opens upwards. This means the range starts from the minimum point on the parabola (the vertex) and goes to positive infinity. In this case, the minimum \( y \)-value occurs at the vertex, yielding a range of \([-\frac{7}{2}, \infty)\).

Parabola

A parabola is the U-shaped curve we get when we graph a quadratic function. It's important to note how the parabola opens. If the coefficient \( a \) from the quadratic function \( f(x) = ax^2 + bx + c \) is positive, the parabola opens upwards, and if \( a \) is negative, it opens downwards.

Some key features of a parabola include:

• The vertex - the highest or lowest point on the parabola.
• The axis of symmetry - a vertical line that passes through the vertex and divides the parabola into two symmetrical halves.
• The direction of opening (upwards or downwards), determined by the sign of \( a \).

Understanding these features helps in analyzing how the parabola behaves across its domain.

Vertex of Quadratic Function

The **vertex** of a quadratic function is a critical point that represents the peak or trough of the parabola. It is the point at which the function reaches its maximum or minimum value. For a quadratic function \( f(x) = ax^2 + bx + c \), the **x-coordinate** of the vertex can be found using the formula \( x = -\frac{b}{2a} \).

In the example function \( f(x) = 2x^2 + 6x - 7 \), we find the x-coordinate of the vertex by substituting \( b = 6 \) and \( a = 2 \) into the formula, resulting in \( x = -\frac{6}{4} = -\frac{3}{2} \). To find the **y-coordinate**, substitute \( x = -\frac{3}{2} \) back into the original equation, giving us a vertex at \( \left(-\frac{3}{2}, -\frac{7}{2}\right) \).

The vertex not only provides valuable information about the graph but also helps us determine the minimum or maximum value of the function, which is crucial for understanding the range.

Real Numbers

The concept of **real numbers** is fundamental in understanding functions, especially when considering the domain and range. Real numbers include all the numbers on the number line, encompassing both rational numbers, like fractions and integers, and irrational numbers, such as \( \sqrt{2} \) or \( \pi \).

In the context of quadratic functions, the domain is always all real numbers. This means that no matter what value of \( x \) you choose, whether it's positive, negative, fractional, or irrational, it can be plugged into the function without issue. This is why the domain is expressed as \((-\infty, \infty)\).

Understanding real numbers is essential when graphing functions like parabolas and calculating the range, since both rely on the capacity to handle any value within the real number line.