Functions $f$ and $g$ are defined by: * $f(x) = 2 ext{sin } x$, where $0 < x < \frac{\pi}{2}$ * $g(x) = \text{z} x$, where $0 < x < \frac{\pi}{4}$ a) (i) Evaluate $f(g(\frac{\pi}{6}))$ - Scottish Highers Maths - Question 13 - 2023

We know from the definitions that:

$g(x) = z x$

Thus, $f(g(x)) = f(z x)$

Substituting in the function $f(x)$:

$f(g(x)) = f(z x) = 2 \text{sin}(z x)$

Therefore, the expression for $f(g(x))$ is:

$f(g(x)) = 2 \text{sin}(z x)$

We have:

$f(p) = 2 \text{sin}(p) = \frac{1}{3}$

Rearranging gives:

$\text{sin}(p) = \frac{1}{6}$

From part (b)(ii), we have:

$g(p) = z p$

Thus, $f(g(p)) = f(z p) = 2 \text{sin}(z p)$

Substituting the value of $p$:

$f(g(p)) = 2 \text{sin}(z \cdot \frac{1}{6})$

This can be evaluated based on the required value of $z$.