Question

What is the inverse of f(x) = sqrt(2x - 3) for x >= 3/2; f ^ - 1 * (x) = 1/2 * (x ^ 2 + 3); x >= 0; f ^ - 1 * (x) = - 1/2 * (x ^ 2 + 3) , x <= 0; f ^ - 1 * (x) = 1/2 * (x ^ 2 + 3) , x < 0; f ^ - 1 * (x) = - 1/2 * (x ^ 2 + 3) , x >= 0

Answer 1

The inverse of the function f(x) = sqrt(2x - 3) can be found by interchanging x and y and then solving for y. The inverse function has multiple pieces depending on the value of x. The inverse of f(x) = sqrt(2x - 3) is:
`y = 1/2 * (x^2 + 3) for x > = 3/2`

`y = -1/2 * (x^2 + 3) for x < = 3/2`

The inverse of the function f(x) = sqrt(2x - 3) can be found by interchanging x and y and then solving for y. To do this, we start by setting y = sqrt(2x - 3) and then swap x and y to get x = sqrt(2y - 3). We can now solve for y by squaring both sides of the equation:

`x^2 = 2y - 3`

`2y = x^2 + 3`

`y = 1/2 * (x^2 + 3) for x > = 3/2`

`y = -1/2 * (x^2 + 3) for x < = 3/2`

To summarize, the inverse of f(x) = sqrt(2x - 3) is:

`y = 1/2 * (x^2 + 3) for x > = 3/2`

`y = -1/2 * (x^2 + 3) for x < = 3/2`