Question

What are invertible functions? plz explain.

Answer 1

See explanation...

Explanation:

There are at least two things implied by the assertion:

"`f(x)` is invertible function `AA x in [1, 5]`"

• `f(x)` is a function with domain including all of the real interval `[1, 5]`

• For any `x_1, x_2 in [1, 5]` we have `f(x_1) = f(x_2) => x_1 = x_2`. In other words, `f(x)` is one to one.

Considered as a function on the domain `[1, 5]`, there is an inverse function `g(x)` on the range of `f(x)`, such that:

`g(f(x)) = x" "` for all `x in [1, 5]`

`f(g(x)) = x" "` for all `x in f("["1, 5"]")`

We are not told anything about the behaviour of `f(x)` outside of this interval. For example, it could be undefined or not one to one.

For the purposes of the rest of the question we do not care.

Given:

`g(3) = 1` and `g(6) = 5`

we can deduce:

`f(1) = 3` and `f(5) = 6`

If the question is correct - which I am not sure - then we can find which multiple choice option is correct by using a linear function for `f(x)` satisfying the conditions.

Let:

`f(x) = 3/4(x-1)+3 = 3/4x+9/4`

`g(x) = 4/3(x-3)+1 = 4/3x-3`

Integrate each one the given intervals and add to find the answer.

I think the question is flawed in at least a couple of ways:

• The integration limits for `g(x)` should be `color(red)(3)` and `6`, not `5` and `6`.

• Some extra conditions on the functions are required to make them integrable. For example, we could specify that `f(x)` is continuous.

With the question corrected, the answer I would expect would be the sum of the areas of three rectangles:

• Vertices `(1, 0), (5, 0), (5, 3), (1, 3)` with area `4*3 = 12`

• Vertices `(1, 3), (5, 3), (5, 6), (1, 6)` with area `4*3 = 12`

• Vertices `(0, 3), (1, 3), (1, 6), (0, 6)` with area `1*3 = 3`

So, total `12+12+3 = 27`, i.e. option 2)