If $f\left(\frac{x}{y}\right)=\frac{f\left(x\right)}{f\left(y\right)}$ and $f\left(5\right)=4$, then $f\left(10\right)-f(-10)$ is equal to

The correct answer is 15.9375.

We are given two pieces of information:

(i) The functional equation: $f\left(\frac{x}{y}\right)=\frac{f\left(x\right)}{f\left(y\right)}$
(ii) $f\left(5\right)=4$

Step 1 — Find f(0).

Substitute $x=y$ into the functional equation:

$f\left(\frac{x}{x}\right)=\frac{f\left(x\right)}{f\left(x\right)}$

$f\left(0\right)=1$

Step 2 — Find the relationship between f(−y) and f(y).

Now substitute $x=0$ into the functional equation:

$f\left(\frac{0}{y}\right)=\frac{f\left(0\right)}{f\left(y\right)}$

$f(-y)=\frac{1}{f\left(y\right)}$

This is a key result: for any value y, the function evaluated at −y is the reciprocal of the function evaluated at y.

Step 3 — Find f(−5).

Using the result from Step 2 with $y=5$:

$f(-5)=\frac{1}{f\left(5\right)}=\frac{1}{4}$

Step 4 — Find f(10).

We substitute $x=5$ and $y=-5$ into the functional equation:

$f\left(\frac{5}{-5}\right)=\frac{f\left(5\right)}{f(-5)}$

$f\left(10\right)=\frac{4}{\frac{1}{4}}=4\times 4=16$

Step 5 — Find f(−10).

Using the reciprocal property from Step 2 with $y=10$:

$f(-10)=\frac{1}{f\left(10\right)}=\frac{1}{16}$

Step 6 — Compute f(10) − f(−10).

$f\left(10\right)-f(-10)=16-\frac{1}{16}$

$=\frac{256}{16}-\frac{1}{16}=\frac{256-1}{16}=\frac{255}{16}$

$=15.9375$

Therefore, $f\left(10\right)-f(-10)=15.9375$.