Problem 10 - Beginning and Intermediate Algebra

Question

For each pair of functions, find a) \(\left(\frac{f}{g}\right)(x)\) and b \(\left(\frac{f}{g}\right)(-2)\) Identify any values that are not in the domain of \(\left(\frac{f}{g}\right)(x)\). $$f(x)=3 x-8, g(x)=x-1$$

Step-by-Step Solution

Answer

a) \(\left(\frac{f}{g}\right)(x) = \frac{3x-8}{x-1}\), b) \(\left(\frac{f}{g}\right)(-2) = \frac{14}{3}\), and the value 1 is not in the domain of the quotient function.

Step 1: Calculate the Quotient Function \(\left(\frac{f}{g}\right)(x)\)

To find the quotient function, divide function f(x) by function g(x): \[\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{3x-8}{x-1}.\]

Step 2: Evaluate the Quotient Function at x = -2

Now, let's find the value of the quotient function at x = -2: \[\left(\frac{f}{g}\right)(-2) = \frac{3(-2)-8}{(-2)-1} = \frac{-14}{-3} = \frac{14}{3}.\]

Step 3: Find Values Not in the Domain of the Quotient Function

The quotient function's domain consists of all the real numbers except the values that make the denominator zero. As we're dividing by g(x), we need to find the x value that makes g(x) = 0: \[x - 1 = 0\] \[x = 1\] Thus, the value 1 is not in the domain of the quotient function \(\left(\frac{f}{g}\right)(x)\). In conclusion, a) the quotient function \(\left(\frac{f}{g}\right)(x)\) is \(\frac{3x-8}{x-1}\), and b) at x = -2, the value of the quotient function is \(\frac{14}{3}\). Moreover, the value 1 is not in the domain of the quotient function.

Key Concepts

Algebraic Functions

Algebraic functions involve mathematical expressions that are composed using the basic operations of addition, subtraction, multiplication, and division, as well as roots. These functions can often be expressed in the form of a polynomial or a rational function. In the given exercise, both functions \(f(x) = 3x - 8\) and \(g(x) = x - 1\) are polynomial functions, which are a type of algebraic function.

• Polynomial functions are composed of terms that are non-negative integer powers of \(x\).
• Rational functions, like the quotient of \(f\) and \(g\), are ratios of two polynomials.

Understanding these basic types helps in identifying how functions can be manipulated through various algebraic operations, such as addition, subtraction, and importantly, division.

Domain of a Function

The domain of a function is the complete set of possible values of the independent variable, typically \(x\), for which the function is defined. In simple words, it is the list of all the \(x\)-values you can plug into a function without causing issues like division by zero or taking the square root of a negative number.
In our example, we calculate the domain of the function \(\left(\frac{f}{g}\right)(x) = \frac{3x-8}{x-1}\).

• The primary concern when finding the domain is to ensure that the denominator is not zero, because division by zero is undefined.
• Set the denominator equal to zero and solve for \(x\): \(x - 1 = 0\), which gives \(x = 1\).
• Thus, the domain includes all real numbers except \(x = 1\).

This restriction means we should exclude \(x = 1\) from the domain to keep the mathematical operation valid.

Function Evaluation

Evaluating a function means determining what the function equals at a specific input value. When functions involve expressions, you substitute the given value into the expression wherever the variable appears.
Here’s how we evaluate the quotient function \(\left(\frac{f}{g}\right)(x)\) at \(x = -2\):

• Substitute \(x = -2\) into the quotient function \(\left(\frac{f}{g}\right)(x) = \frac{3x-8}{x-1}\).
• Calculate the numerator: \(3(-2) - 8 = -6 - 8 = -14\).
• Calculate the denominator: \(-2 - 1 = -3\).
• The value is then \(\frac{-14}{-3} = \frac{14}{3}\).

Function evaluation is a crucial skill because it provides specific values of functions at given points, essential in various fields of application including science and engineering.

Division of Functions

Dividing two functions involves creating a new function that represents the ratio of the original functions. Specifically, for functions \(f(x)\) and \(g(x)\), their quotient is \(\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}\).

• Make sure that \(g(x) eq 0\) to avoid undefined values.
• The division creates a rational function if \(f(x)\) and \(g(x)\) are polynomials.

In our example with \(f(x) = 3x - 8\) and \(g(x) = x - 1\), dividing these gives \(\left(\frac{f}{g}\right)(x) = \frac{3x-8}{x-1}\).
This operation not only creates a new function but also requires careful consideration of the domain due to potential zero values in the denominator. Such division is fundamental in calculus and algebra, allowing for transformations and simplifications in various mathematical problems.