Question

Determine if the two functions are inverses of one another.

f(x) = -2x + 1

g(x) = x - 1

Options:

1. Yes, both functions are inverses of each other.
2. No, these functions fail to be inverses of each other.

Answer 1

After computing the composite functions f(g(x)) and g(f(x)), neither composition equates to x, proving that f(x) = -2x + 1 and g(x) = x - 1 are not inverses of each other.

Step-by-step explanation:

Two functions are inverses of each other if the composition of one with the other returns the identity function for every element in their domain. Specifically, if we have functions f(x) and g(x), they are inverses if f(g(x)) = x and g(f(x)) = x for all x in the domain of both functions. To determine if f(x) = -2x + 1 and g(x) = x - 1 are inverses of one another, we should find both composite functions.

First, we calculate f(g(x)):

f(g(x)) = f(x - 1) = -2(x - 1) + 1, which simplifies to -2x + 2 + 1 or -2x + 3.

Second, we calculate g(f(x)):

g(f(x)) = g(-2x + 1) = (-2x + 1) - 1 which simplifies to -2x.

As we can see, neither f(g(x)) nor g(f(x)) equals x.

Hence, these functions are not inverses of each other. Therefore, the correct answer is 2. No, these functions fail to be inverses of each other.