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).

1 Answer
May 8, 2018

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 .

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