Question

Given f(x)=(x+1)/(-x-2) and g(x)=(-x+1)/(2x-4), the domain of (f(x))/(g(x))

Answer 1

To determine the domain of the composite function (f(x))/(g(x)), we exclude x-values that result in zero denominators in the original functions f(x) and g(x), which are x = -2 and x = 2, respectively. Thus, the domain of (f(x))/(g(x)) is all real numbers except x = -2 and x = 2.

Step-by-step explanation:

Finding the Domain of the Function f(x)/g(x)

To find the domain of the composite function (f(x))/(g(x)), we need to identify the values of x for which both f(x) and g(x) are defined and g(x) is not equal to zero, since division by zero is undefined. The original functions are f(x)=(x+1)/(-x-2) and g(x)=(-x+1)/(2x-4).

The domain of f(x) is all real numbers except x = -2, as that makes the denominator zero. Similarly, the domain of g(x) is all real numbers except x = 2, because substituting x = 2 in the denominator 2x - 4 gives zero. Additionally, since g(x) will be the denominator of our new function, we must also ensure that g(x) ≠ 0.

Combining these restrictions, the domain of (f(x))/(g(x)) is all real numbers except x = -2 and x = 2.

Answer 2

The domain of
`\((f(x))/(g(x))\)` is all real numbers except
`\(x = -2\) and \(x = 2\)`because these values make the denominator of
`\(g(x)\)` equal to zero, resulting in division by zero, which is undefined.

Step-by-step explanation:

To find the domain of
`\((f(x))/(g(x))\)`, we need to identify any values of
`\(x\)` that would make the denominator of
`\(g(x)\)` equal to zero, as division by zero is undefined in mathematics. The denominator of
`\(g(x)\) is \(2x - 4\), which becomes zero when \(x = 2\) since \(2 * 2 - 4 = 0\). Therefore, \(x = 2\)`is a value that causes the expression to be undefined. Similarly, for the function
`\(g(x)\)`, if we solve for \(x\) in
`\(2x - 4 = 0\), we find \(x = 2\),`indicating that
`\(x = 2\)` is a value excluded from the domain of
`\(g(x)\)` to avoid division by zero.

Hence, the domain of
`\((f(x))/(g(x))\)` comprises all real numbers except
`\(x = -2\) and \(x = 2\)` since these values would cause the denominator of
`\(g(x)\)`to become zero. Any other real number for
`\(x\)` would allow the expression
`\((f(x))/(g(x))\)` to be defined and valid. Therefore, the domain of the quotient function
`\((f(x))/(g(x))\)` encompasses the set of all real numbers except for the values that result in the denominators becoming zero.