Find f/g and determine the domain for each function. f(x) = 2x + 3, g(x) = x − 1

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First, write down the given functions: \(f(x) = 2x + 3\) and \(g(x) = x - 1\).

To find \(\frac{f}{g}\), form the quotient of the two functions: \(\frac{f}{g} = \frac{2x + 3}{x - 1}\).

Next, determine the domain of \(\frac{f}{g}\). The domain includes all real numbers except where the denominator is zero, because division by zero is undefined.

Set the denominator equal to zero and solve for \(x\): \(x - 1 = 0\) which gives \(x = 1\). This value must be excluded from the domain.

Therefore, the domain of \(\frac{f}{g}\) is all real numbers \(x\) such that \(x \neq 1\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Function Division (f/g)

Dividing two functions f and g, denoted as (f/g)(x), means creating a new function by dividing the output of f(x) by g(x). This is expressed as (f/g)(x) = f(x) / g(x). It is important to perform the division carefully and simplify if possible.

Domain of a Function

The domain of a function is the set of all input values (x) for which the function is defined. When dividing functions, the domain excludes any x-values that make the denominator zero, since division by zero is undefined.

Linear Functions

Both f(x) = 2x + 3 and g(x) = x − 1 are linear functions, meaning their graphs are straight lines. Understanding their behavior helps in identifying values that affect the domain, especially where g(x) equals zero, which must be excluded.

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