Problem 7 - Precalculus : Building Concepts and Connections

Question

In Exercises \(7-16,\) for the given functions \(f\) and \(g,\) find each composite function and identify its domain. (a) \((f+g)(x)\) (b) \((f-g)(x)\) (c) \((f g)(x)\) (d) \(\left(\frac{f}{g}\right)(x)\) $$f(x)=3 x-5 ; g(x)=-x+3$$

Step-by-Step Solution

Answer

\((f+g)(x) = 2x - 2, (f-g)(x) = 4x - 8, (f g)(x) = -3x^2+14x-15, (f/g)(x) = (3x-5)/(-x+3)\) with domains respectively all real numbers, all real numbers, all real numbers, and all real numbers except x = 3.

Step 1: Compute (f+g)(x)

To compute \((f+g)(x)\), add the defined functions, \(f(x) = 3x-5\) and \(g(x) = -x+3\). Perform the operation, \(f(x) + g(x) = (3x - 5) + (-x + 3) = 2x - 2\). The domain of this function is all real numbers as there are no values of x that would make this function undefined.

Step 2: Compute (f-g)(x)

To compute \((f-g)(x)\), subtract g(x) from f(x). Perform the operation, \(f(x) - g(x) = (3x - 5) - (-x + 3) = 4x - 8\). The domain of this function is all real numbers as there is no value of x that would make this function undefined.

Step 3: Compute (fg)(x)

To compute \(f(x)g(x)\), multiply the functions f(x) and g(x), resulting in \((3x-5)(-x+3) = -3x^2+9x+5x-15 = -3x^2+14x-15\). The domain of this function is all real numbers, as there is no value of x that would make this function undefined.

Step 4: Compute \((f/g)(x)\)

To compute \((f/g)(x)\), divide the function f(x) by the function g(x). This yields \( (f/g)(x) = (3x-5)/(-x+3)\). Here the domain is all real numbers except for the value that makes g(x) equal to zero (i.e., the denominator cannot be zero since division by zero is undefined). Solve for -x+3 = 0, giving x = 3. So, the domain of this function is all real numbers except x = 3.

Key Concepts

Domain of a Function

Understanding the domain of a function is crucial when working with mathematical expressions. The **domain** of a function refers to all the values that the input, or the variable (usually represented as \(x\)), can take so that the function is defined.

For simple functions such as linear functions like \(f(x) = 3x - 5\) and \(g(x) = -x + 3\), the domain is all real numbers. This is because any real number substituted for \(x\) will produce a legitimate output without causing any mathematical issues.

However, operations such as division can complicate the domain. For instance, with a function like \((f/g)(x) = \frac{3x - 5}{-x + 3}\), we need to be careful. Division by zero is undefined in mathematics, so we need to exclude any \(x\) values that would make the denominator zero. Here, solving \(-x + 3 = 0\) shows that when \(x = 3\), the function becomes undefined. Thus, the domain of this function is all real numbers except \(x = 3\).

Paying attention to the domain helps avoid undefined expressions and ensures accurate mathematical results.

Operations on Functions

Operations on functions involve executing basic mathematical operations such as addition, subtraction, multiplication, and division on two or more functions. These operations allow us to create new functions based on existing ones.

• **Addition**: The sum of two functions, \((f+g)(x)\), is found by adding their respective functions. For example, using \(f(x) = 3x - 5\) and \(g(x) = -x + 3\), the result is \((f+g)(x) = (3x - 5) + (-x + 3) = 2x - 2\).
• **Subtraction**: Subtracting one function from another, \((f-g)(x)\), involves subtracting their expressions. For \(f(x) - g(x)\), this results in \(4x - 8\).
• **Multiplication**: To multiply functions \((fg)(x)\), you multiply the expressions of \(f(x)\) and \(g(x)\). For example, \((3x - 5)(-x + 3) = -3x^2 + 14x - 15\).
• **Division**: The quotient of two functions, \((f/g)(x)\), requires dividing \(f(x)\) by \(g(x)\), being mindful that \(g(x)\) must not be zero to avoid undefined expressions.

Each of these operations may affect the domain of the resulting function, especially division can lead to restrictions on the domain by making certain values undefined.

Function Composition

Function composition involves creating a new function by applying one function to the result of another, denoted as \((f \circ g)(x)\). This results in applying \(g(x)\) first and then applying \(f\) to the result of \(g(x)\).

To compose functions, follow these steps:

• **Evaluate** \(g(x)\) first for any given \(x\), producing an intermediate result.
• **Substitute** this intermediate result into the function \(f(x)\).

For example, if \(g(x) = -x + 3\) and \(f(x) = 3x - 5\), the composition \((f \circ g)(x)\) would be evaluated as \(f(g(x)) = f(-x + 3)\). Substitute \(-x + 3\) into \(f(x)\), resulting in \(3(-x + 3) - 5 = -3x + 9 - 5 = -3x + 4\).

When considering function composition, it's essential to understand the domains of the original functions. The input to \(f\) must be within the domain of \(g\). Thus, the domains of combined functions should be checked to avoid undefined results.