Question

If exactly two different linear functions, f and g, satisfy f(f(x)) = g(g(x)) = 4x + 3, what is the product of f(1) and g(1)?

Answer 1

`f(1)g(1) = -15`

Explanation:

The effect of applying `f(x)` twice is roughly to multiply `x` by `4` - especially for large values of `x`.

Since it is a linear function, it must take the form:

`f(x) = 2x+c`

or:

`f(x) = -2x+c`

Note that:

`2(2x+c)+c = 4x+3c`

So `f(x)` could be `2x+1`

Alternatively:

`-2(-2x+c)+c = 4x-c`

So `f(x)` could be `-2x-3`

These are the only two possibilities, so let:

`f(x) = 2x+1`

`g(x) = -2x-3`

Then:

`f(1)g(1) = (2(1)+1)(-2(1)-3) = 3(-5) = -15`