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)?

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Answer 1 · 2017-12-21T00:13:04

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

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

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

$f (x) = 2 x + c$ $f(x) = 2x+c$

or:

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

Note that:

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

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

Alternatively:

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

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

These are the only two possibilities, so let:

$f (x) = 2 x + 1$ $f(x) = 2x+1$

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

Then:

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