Problem 65 - Elementary and Intermediate Algebra

Question

Let \(f(x)=3 x-2\) and \(g(x)=2 x^{2}+1 .\) Find each function and give its domain. $$ f-g $$

Step-by-Step Solution

Answer

\((f-g)(x) = -2x^2 + 3x - 3\) with domain \(\mathbb{R}\).

Step 1: Define f(x) and g(x)

We are given the functions \(f(x) = 3x - 2\) and \(g(x) = 2x^2 + 1\). These are both polynomial functions. The domain of any polynomial function is all real numbers, \(\mathbb{R}\).

Step 2: Find the Expression for (f-g)(x)

To find \((f-g)(x)\), we subtract \(g(x)\) from \(f(x)\). Thus, \((f-g)(x) = f(x) - g(x) = (3x - 2) - (2x^2 + 1)\).

Step 3: Simplify the Expression

Simplify the expression from Step 2: \[(f-g)(x) = 3x - 2 - 2x^2 - 1 = -2x^2 + 3x - 3\].

Step 4: Determine the Domain of (f-g)(x)

Since \((f-g)(x) = -2x^2 + 3x - 3\) is a quadratic expression, its domain is the same as that of any polynomial, which is all real numbers, \(\mathbb{R}\).

Key Concepts

Function Operations

Function operations allow us to add, subtract, multiply or divide functions. When dealing with polynomial functions like in our example, these operations can be quite straightforward. Suppose we have two functions, \( f(x) \) and \( g(x) \). The operations are defined as follows:

• Addition: \( (f+g)(x) = f(x) + g(x) \)
• Subtraction: \( (f-g)(x) = f(x) - g(x) \)
• Multiplication: \( (f \cdot g)(x) = f(x) \cdot g(x) \)
• Division: \( \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} \), \( g(x) eq 0 \)

In the example exercise, we performed the subtraction of two functions \( f(x) = 3x - 2 \) and \( g(x) = 2x^2 + 1 \). We found \((f-g)(x)\) by computing \((3x - 2) - (2x^2 + 1)\), which simplifies to \(-2x^2 + 3x - 3\). This new expression combines parts of each original function in a way that showcases differences between them.

Domain of a Function

The domain of a function is an important concept as it tells us all the possible input values \(x\) for which the function is defined. For polynomial functions, like the ones in this exercise, the domain is all real numbers, symbolized as \(\mathbb{R}\). This means you can plug any real number into the polynomial function, and it will provide a valid output.

For the functions \( f(x) = 3x - 2 \) and \( g(x) = 2x^2 + 1 \), we do not have any restrictions based on the operations of addition, subtraction, multiplication, or division. Thus, when we look at \((f-g)(x) = -2x^2 + 3x - 3\), it remains that the domain is all real numbers \( \mathbb{R} \), because it is still a polynomial function. This consistent domain makes polynomial function operations very convenient to work with.

Quadratic Expression

Understanding quadratic expressions is crucial in recognizing different polynomial types. A quadratic expression features a term with \(x^2\) as its highest power. Its standard form is \(ax^2 + bx + c\), where the coefficients \(a\), \(b\), and \(c\) are real numbers, and importantly, \(a eq 0\).

In our example, upon performing \(f-g\), we get the quadratic expression \(-2x^2 + 3x - 3\). Here, we observe:

• \(a = -2\): The coefficient of \(x^2\)
• \(b = 3\): The coefficient of \(x\)
• \(c = -3\): The constant term

These terms combine to form a parabola when this expression is plotted on a graph. The shape and direction of the parabola are determined by the sign and magnitude of \(a\). Since \(a\) is negative, the parabola opens downwards. Quadratic expressions are fundamental in mathematics, appearing in various real-world problems and scenarios.