Question

# Given that f(x, y, z) = xy + z, x = s^2, y = st, z = t^2, find the composite function.

Composite functions
Given that
$$\displaystyle{f{{\left({x},{y},{z}\right)}}}={x}{y}+{z},$$
$$\displaystyle{x}={s}^{{2}},$$
$$\displaystyle{y}={s}{t},$$
$$\displaystyle{z}={t}^{{2}},$$
find the composite function.

2021-03-10
$$\displaystyle{f{{\left({x},{y},{z}\right)}}}={x}{y}+{z}$$
Substitute $$\displaystyle{x}={s}^{{2}}$$, $$\displaystyle{y}={s}{t}$$, and $$\displaystyle{z}={t}^{{2}}$$
$$\displaystyle{f{{\left({x},{y},{z}\right)}}}={\left({s}^{{2}}\right)}{\left({s}{t}\right)}+{t}^{{2}}$$
$$\displaystyle{f{{\left({x},{y},{z}\right)}}}={\left({s}^{{2}}\cdot{s}^{{1}}{t}\right)}+{t}^{{2}}$$
$$\displaystyle{f{{\left({x},{y},{z}\right)}}}={\left({s}^{{2}}+{1}\cdot{t}\right)}+{t}^{{2}}$$ [because $$\displaystyle{a}^{{n}}\cdot{a}^{{m}}={a}^{{{n}+{m}}}$$]
$$\displaystyle{f{{\left({x},{y},{z}\right)}}}={s}^{{3}}{t}+{t}^{{2}}$$