# 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)?

Dec 21, 2017

$f \left(1\right) g \left(1\right) = - 15$

#### Explanation:

The effect of applying $f \left(x\right)$ 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:

$f \left(x\right) = 2 x + c$

or:

$f \left(x\right) = - 2 x + c$

Note that:

$2 \left(2 x + c\right) + c = 4 x + 3 c$

So $f \left(x\right)$ could be $2 x + 1$

Alternatively:

$- 2 \left(- 2 x + c\right) + c = 4 x - c$

So $f \left(x\right)$ could be $- 2 x - 3$

These are the only two possibilities, so let:

$f \left(x\right) = 2 x + 1$

$g \left(x\right) = - 2 x - 3$

Then:

$f \left(1\right) g \left(1\right) = \left(2 \left(1\right) + 1\right) \left(- 2 \left(1\right) - 3\right) = 3 \left(- 5\right) = - 15$