Question

State the smallest value of k for which g has an inverse?

The function g is such that `g(x) = 8 - (x - 2)^2,` for `k <= x <= 4,`where k is a constant.
(i) State the smallest value of k for which g has an inverse.
For this value of k ,
(ii) Find an expression for `g^-1(x)`.
(iii) Sketch, on the same diagram, the graphs of `y = g(x) and y = g^-1(x).`

Answer 1

`k=2` and `g^{-1}(y) = 2 + sqrt{8-y}`

Explanation:

Had a nice answer then a browser crash. Let's try again.

`g(x) = 8-(x-2)^2` for `k le x le 4`

Here's the graph:

graph{8-(x-2)^2 [-5.71, 14.29, -02.272, 9.28]}

The inverse exists over a domain of `g` where `g(x)` doesn't have the same value for two different values of `x`. Less than 4 means we can go to the vertex, clearly from the expression or the graph at `x=2.`

So for (i) we get `k=2`.

Now we seek `g^{-1}(x)` for `2 le x le 4.`

`g(x) = y = 8 -(x-2)^2`

`(x-2)^2 = 8-y`

We're interested in the side of the equation where `x ge 2.` That means `x-2 ge 0` so we take the positive square root of both sides:

`x-2 = sqrt{8-y}`

`x = 2 + sqrt{8-y}`

`g^{-1}(y) = 2 + sqrt{8-y} quad`

That's the answer for (ii)

Sketch. We'll go with Alpha .