Problem 32 - Precalculus Essentials

Question

Find \(f+g, f-g,\) fg, and \(\frac{f}{x}\). Determine the domain for each function. $$f(x)=3 x-4, g(x)=x+2$$

Step-by-Step Solution

Answer

\(f+g)(x) = 4x - 2\), domain is all real numbers. \(f-g)(x) = 2x - 6\), domain is all real numbers. \(fg(x) = 3x^2 + 2x - 8\), domain is all real numbers. \(\frac{f}{g} = \frac{(3x - 4)}{(x+2)}\), domain is all real numbers except -2.

Step 1: Adding the functions

To find \(f+g\), simply add the two functions together. \[(f+g)(x) = f(x) + g(x) = (3x - 4) + (x + 2)\] This simplifies to \[(f+g)(x) = 4x - 2\] The domain for this function is all real numbers as it is defined for all real values of x.

Step 2: Subtracting the functions

To find \(f-g\), subtract g(x) from f(x). \[(f-g)(x) = f(x) - g(x) = (3x - 4) - (x + 2)\] This simplifies to \[(f-g)(x) = 2x - 6\] The domain for this function is all real numbers as it is also defined for all real values of x.

Step 3: Multiplying the functions

To find fg, multiply the two functions together. \[fg(x) = f(x) * g(x) = (3x - 4) * (x + 2)\] This simplifies to \[fg(x) = 3x^2 + 2x - 8\] The domain for this function is all real numbers as it is defined for all real values of x.

Step 4: Dividing the functions

To find \(\frac{f(x)}{g(x)}\), divide f(x) by g(x). \[\frac{f(x)}{g(x)} = \frac{(3x - 4)}{(x+2)}\] The domain for this function is a bit different as it is undefined for x = -2 because we do not define division by zero in mathematics. So the domain will be all real numbers except -2.

Key Concepts

Function Addition and Subtraction

The operations of addition and subtraction for functions are analogous to those for regular numbers, but here we're dealing with expressions. Just as with numbers, you can add or subtract function values simply by combining like terms. For example, when you add the functions \( f(x) = 3x - 4 \) and \( g(x) = x + 2 \), you align the like terms and combine them, resulting in \( (f+g)(x) = 4x - 2 \).

Similarly, subtraction operates under the same principle where \( (f-g)(x) \) means you subtract \( g(x) \) from \( f(x) \), and therefore \( (f-g)(x) = 2x - 6 \). These operations are straightforward and the resulting functions often retain the same domain as the original functions unless subtraction leads to terms like \( \frac{1}{x} \) which introduces new restrictions.

Function Multiplication

Multiplying functions incorporates the familiar distributive property, which you use when multiplying binomials. In function multiplication, you multiply each term of one function by each term of the other. With \( f(x) = 3x - 4 \) and \( g(x) = x + 2 \), this gives us \( fg(x) = (3x - 4)(x + 2) \), which simplifies to \( fg(x) = 3x^2 + 6x - 8 \).

The domain of the resulting function is the set of all input values for which both original functions are defined. In the case of \( f \) and \( g \) provided, since both functions are defined for all real numbers, the domain for their product is all real numbers too.

Function Division

Division of functions is slightly more complex because it involves a restriction: you cannot divide by zero. When dividing \( f(x) \) by \( g(x) \), denoted as \( \frac{f(x)}{g(x)} \), you form the quotient by placing one function over the other, much like a fraction. In our case, we have \( \frac{f(x)}{g(x)} = \frac{3x - 4}{x + 2} \).

While the numerator does not impose new domain restrictions, we must exclude any real number from the domain that makes the denominator zero, since division by zero is undefined. Thus, the domain of \( \frac{f(x)}{g(x)} \) is all real numbers except for \( x = -2 \), the value that would make the denominator zero. Understanding this domain exclusion is critical for grasping the essence of function division.

Domain of a Function

The domain of a function is the complete set of possible values of the independent variable which will output real numbers from a particular function. In simpler terms, it represents all the values that you can plug into a function and get a real number in return.

For polynomial functions, such as the ones we encountered with \( f(x) = 3x - 4 \) and \( g(x) = x + 2 \), the domain is typically all real numbers because you can insert any real number into a polynomial and you will receive a real number back. However, when functions include denominators or square roots, you need to be more cautious because the domain will exclude any numbers that result in division by zero or taking the square root of a negative number.

Understanding the domain is crucial before performing operations on functions since it impacts the validity of the resulting function values.