For the pair of functions defined, find (f/g)(x).Give the domain of each. See Example 2. ƒ(x)=3x+4, g(x)=2x-8

Verified step by step guidance

1. Step 1: Understand the problem. We need to find the function (f/g)(x), which is the division of the function f(x) by g(x).
2. Step 2: Write the expression for (f/g)(x). This is given by (f/g)(x) = f(x) / g(x) = (3x + 4) / (2x - 8).
3. Step 3: Determine the domain of (f/g)(x). The domain of a function is the set of all possible input values (x-values) that will not cause the function to be undefined.
4. Step 4: Identify restrictions on the domain. For (f/g)(x), the function is undefined when the denominator is zero. Set the denominator equal to zero and solve for x: 2x - 8 = 0.
5. Step 5: Solve the equation 2x - 8 = 0 to find the x-value that makes the denominator zero. Exclude this x-value from the domain of (f/g)(x). The domain of f(x) is all real numbers, and the domain of g(x) is all real numbers except the value that makes the denominator zero.

Key Concepts

Function Division

Function division involves creating a new function by dividing one function by another. In this case, (f/g)(x) is defined as f(x) divided by g(x), which means you will compute (3x + 4) / (2x - 8). Understanding how to manipulate and simplify rational functions is crucial for solving such problems.

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For the function (f/g)(x), the domain must exclude any values that make the denominator zero, as division by zero is undefined. In this case, you need to find the value of x that makes g(x) = 0.

Rational Functions

Rational functions are functions that can be expressed as the ratio of two polynomials. They often exhibit specific behaviors, such as asymptotes and discontinuities, based on the values of the numerator and denominator. Understanding the characteristics of rational functions helps in analyzing their graphs and determining their domains.