If $f(x + y) = f (x) f (y)$ and $f(5) = 4$, then $f (10) - f (-10)$ is equal to

We have a functional equation: $f(x + y) = f(x)f(y)$ and we know that $f(5) = 4$.

This functional equation tells us that when we add inputs, we multiply outputs.

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Since $10 = 5 + 5$, we can use our functional equation:

$f(10) = f(5 + 5) = f(5) \times f(5) = f(5)^2$

Substituting $f(5) = 4$:

$f(10) = 4^2 = 16$

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Before finding $f(-10)$, we need to determine $f(0)$.

Using the functional equation with $x = y = 0$:

$f(0) = f(0 + 0) = f(0) \times f(0) = f(0)^2$

This gives us: $f(0) = f(0)^2$

$f(0) - f(0)^2 = 0$

$f(0)(1 - f(0)) = 0$

So either $f(0) = 0$ or $f(0) = 1$.

If $f(0) = 0$, then $f(5) = f(5 + 0) = f(5) \times f(0) = f(5) \times 0 = 0$

But we're told $f(5) = 4

eq 0$, so this is impossible.

Therefore: $f(0) = 1$

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Using the functional equation with $x = 5$ and $y = -5$:

$f(0) = f(5 + (-5)) = f(5) \times f(-5)$

Since $f(0) = 1$ and $f(5) = 4$:

$1 = 4 \times f(-5)$

$f(-5) = \dfrac{1}{4}$

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Since $-10 = (-5) + (-5)$:

$f(-10) = f(-5 + (-5)) = f(-5) \times f(-5) = f(-5)^2$

Substituting $f(-5) = \dfrac{1}{4}$:

$f(-10) = \left(\frac{1}{4}\right)^2 = \dfrac{1}{16}$

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$f(10) - f(-10) = 16 - \dfrac{1}{16}$

Converting to decimal:

$16 - \dfrac{1}{16} = 16 - 0.0625 = 15.9375$

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In functional equations like $f(x + y) = f(x)f(y)$, negative inputs give reciprocal outputs. Notice how $f(-5) = \dfrac{1}{4}$ while $f(5) = 4$, and $f(-10) = \dfrac{1}{16}$ while $f(10) = 16$.

Answer: 15.9375