Problem 6 - College Algebra and Calculus: An Applied Approach

Question

Find (a) \((f+g)(x)\), (b) \((f-g)(x)\), (c) \((f g)(x)\), and (d) \((f / g)(x)\). What is the domain of \(f / g\) ? \(f(x)=2 x-3, g(x)=1-x\)

Step-by-Step Solution

Answer

(a) (f+g)(x) = x - 2, (b) (f-g)(x) = 3x - 4, (c) (f⋅g)(x) = -3x^2 + 2x - 3, (d) (f/g)(x) = (2x - 3) / (1 - x). The domain of (f/g) is all real numbers except \(1\).\

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

The sum of two functions \(f\) and \(g\) is obtained by adding their function outputs. So, \((f+g)(x) = f(x) + g(x) = (2x - 3) + (1 - x) = x - 2.\

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

The difference between two functions \(f\) and \(g\) is obtained by subtracting the function output of \(g\) from \(f\). So, \((f-g)(x) = f(x) - g(x) = (2x - 3) - (1 - x) = 3x - 4.\

Step 3: Find (f⋅g)(x)

The product of two functions \(f\) and \(g\) is obtained by multiplying their function outputs. So, \((f⋅g)(x) = f(x) ⋅ g(x) = (2x - 3) ⋅ (1 - x) = 2x - 3x^2 - 3 + 3x = -3x^2 + 2x - 3.\

Step 4: Find (f/g)(x)

The quotient of two functions \(f\) and \(g\) is obtained by dividing the function output of \(f\) by \(g\). So, \((f/g)(x) = f(x) / g(x) = (2x - 3) / (1 - x).\

Step 5: Find domain of (f/g)(x)

The domain of a function is the set of all possible inputs x. When the function involves division, we need to exclude any values of x that would make the denominator zero. Thus, to find the domain of the function \(f/g\), we have to solve the equation \(g(x) ≠ 0\), which leads to the inequality \(1 - x ≠ 0\). Solving for \(x\) we get \(x ≠ 1\). Therefore, the domain of the function \(f/g\) is all real numbers except \(1\).\

Key Concepts

Domain of a Function

Understanding the domain of a function is crucial when working with functions. The domain is essentially the set of all possible input values (usually denoted as \(x\)) that will yield meaningful or valid output results. In mathematical expressions involving functions, the domain can often be given implicitly through the nature of the expression. For arithmetic operations like addition, subtraction, and multiplication, the domain typically includes all real numbers unless restricted by context.

However, when division is involved, it is vital to ensure that the denominator is not zero. This is because division by zero is undefined in mathematics. To find the domain of a quotient function, you need to identify the values of \(x\) that make the denominator equal to zero and exclude them from the domain. For example, in the expression \((f/g)(x) = (2x - 3)/(1 - x)\), solving \(1 - x eq 0\) gives \(x eq 1\). Thus, \(x = 1\) is excluded from the domain, making the domain all real numbers except \(x = 1\).

When determining domains, always carefully consider operations that involve division or square roots, as these are typical cases where restrictions occur.

Function Addition and Subtraction

Function addition and subtraction are straightforward processes that follow common arithmetic principles. These operations involve combining or separating the outputs of functions pointwise. This means, for any input \(x\), you calculate the function values separately, and then add or subtract those values together.

Let's take two functions, \(f(x) = 2x - 3\) and \(g(x) = 1 - x\). When you add them up to find \((f + g)(x)\), you simply add their outputs:

• \((f+g)(x) = f(x) + g(x) = (2x - 3) + (1 - x) = x - 2\)

Similarly, function subtraction is executed by subtracting the output of one function from the other:

• \((f-g)(x) = f(x) - g(x) = (2x - 3) - (1 - x) = 3x - 4\)

This operation does not typically affect the domain of the functions unless the initial functions have specific restrictions. Therefore, the domain for the result of addition and subtraction remains the same as the common domain of the initial functions.

Function Multiplication and Division

Function multiplication and division are operations where the outputs of two functions are multiplied or divided. For multiplication, this process is usually straightforward. You take each output value and multiply them directly:

• For our given functions, \((f \cdot g)(x) = (2x - 3)(1 - x) = -3x^2 + 2x - 3\)

Multiplication generally does not impose additional restrictions on the domain unless the initial functions have constraints.

Division, however, requires more attention because it involves potential restrictions on the domain. You must avoid dividing by zero, which is why you need to verify that the divisor (the function in the denominator) is not zero for any input values:

• In the function \((f/g)(x) = (2x - 3)/(1 - x)\), ensure that \(1 - x eq 0\). Solving gives \(x eq 1\).

Thus, the domain of the division function excludes \(x=1\). This highlights the importance of checking for zero in the denominator when performing division, ensuring the result is mathematically valid.