Question

Use f(x) = 4x - 3, g(x) = 1 - 5x, and h(x) = x^2 + 3x - 7. For (g/f)(x), are there any x-values that can't be included in the domain? If so, type the value that is restricted from the domain. If there aren't any, type 'none'.

Answer 1

The composite function (g/f)(x), where f(x) = 4x - 3 and g(x) = 1 - 5x, cannot have x = 3/4 in its domain because it would result in division by zero, which is undefined.

Step-by-step explanation:

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

For the composite function (g/f)(x), where f(x) = 4x - 3 and g(x) = 1 - 5x, we must consider the domains of both functions when combining them. We have to ensure that we do not divide by zero when we perform the division necessary for the composite, which is (g/f)(x) = g(x)/f(x). To find the domain of (g/f)(x), we need to determine when f(x) is not equal to 0.

First, let's find the value of x that would make f(x) equal to zero:

1. Set f(x) equal to zero: 0 = 4x - 3
2. Solve for x: x = 3/4

Since f(x) cannot be zero for the division (g/f)(x) to be valid, this means that

x = 3/4

is restricted from the domain of the composite function (g/f)(x).

If we were to include x = 3/4 in the domain, we would be dividing by zero, which is undefined. Therefore, the value that is restricted from the domain of (g/f)(x) is 3/4.