Question

If f(x) = 3x-6 and g(x) = 1/3x+1, then (g(f))^-1 (x) equals.

1-x
1/3(3x-1)
(x+1)
(x-1)

Answer 1

To find the inverse of the composite function (g(f(x))), you first compute g(f(x)) by substituting f(x) into g(x) to get x - 1, and then find the inverse, which results in (g(f))⁻¹(x) = x + 1.

Step-by-step explanation:

The question involves composite functions and finding inverses of functions. To find the inverse of the composite function (g(f(x))) we need to compose g with f first, and then find the inverse of the resulting function. The function f(x) = 3x - 6, and the function g(x) = 1/3x + 1. When composing g(f(x)), we substitute f(x) into g(x), obtaining g(f(x)) = 1/3(3x - 6) + 1. Simplifying this gives g(f(x)) = x - 1. The inverse of this function, (g(f))-1(x), would undo what g(f(x)) does, so we solve for x in the equation y = x - 1. Adding 1 to both sides gives us x = y + 1, or written as the inverse function: (g(f))⁻¹(x) = x + 1.

Answer 2

`(g(f))^-1(x) equals 1/3(3x-1).`

The composition
`(g(f))(x)` simplifies to
`1/3(3x-1)` by substituting the given functions g(x) and f(x) and finding the inverse.

Step-by-step explanation:

The composition (g(f))(x) represents the combination of the functions g and f, denoted as g(f(x)). To find the inverse of this composition, we set the entire expression equal to x and solve for the variable. Let
`y = (g(f))(x)`, then
`g(f(y)) = x.`

Given
`g(x) = 1/3x + 1` and
`f(x) = 3x - 6`, we first find
`g(f(x))`. Substituting f(x) into g(x), we get
`g(f(x)) = g(3x - 6) = 1/3(3x - 6) + 1`. Simplifying this expression yields
`g(f(x)) = x - 1`.

Now, we have y = x - 1. To find the inverse, we switch x and y and solve for y. Swap x and y: x = y - 1. Solving for y, we get y = x + 1.

However, the original question asked for (g(f))^-1(x), so we substitute this expression back into the result for
`y: (g(f))^-1(x) = 1/3(3x - 1)`. Therefore, the final answer is
`1/3(3x - 1)`.