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

Question
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 $$\displaystyle{a}^{{n}}\cdot{a}^{{m}}={a}^{{{n}+{m}}}$$]
$$\displaystyle{f{{\left({x},{y},{z}\right)}}}={s}^{{3}}{t}+{t}^{{2}}$$

### Relevant Questions

Find the following derivatives.
$$\displaystyle{z}_{{{s}}}\ {\quad\text{and}\quad}\ {z}_{{{t}}},\ {w}{h}{e}{r}{e}\ {z}={e}^{{{x}+{y}}},{x}={s}{t},\ {\quad\text{and}\quad}\ {y}={s}+{t}$$
Let X and Y be independent, continuous random variables with the same maginal probability density function, defined as
$$f_{X}(t)=f_{Y}(t)=\begin{cases}\frac{2}{t^{2}},\ t>2\\0,\ otherwise \end{cases}$$
(a)What is the joint probability density function f(x,y)?
(b)Find the probability density of W=XY. Hind: Determine the cdf of Z.
Assume that X and Y are jointly continuous random variables with joint probability density function given by
$$f(x,y)=\begin{cases}\frac{1}{36}(3x-xy+4y)\ if\ 0 < x < 2\ and\ 1 < y < 3\\0\ \ \ \ \ othrewise\end{cases}$$
Find the marginal density functions for X and Y .
Given
$$\displaystyle{f{{\left({x}\right)}}}={x}^{{2}}+{x}+{1}$$
h(x) = 3x + 2,
evaluate the composite function.
Find the first partial derivatives of the function f given below.
$$\displaystyle{f{{\left({x},{y}\right)}}}={5}{e}^{{{x}{y}+{2}}}$$
Let f(x) = $$\displaystyle{4}{x}^{{2}}–{6}$$ and $$\displaystyle{g{{\left({x}\right)}}}={x}–{2}.$$
(a) Find the composite function $$\displaystyle{\left({f}\circ{g}\right)}{\left({x}\right)}$$ and simplify. Show work.
(b) Find $$\displaystyle{\left({f}\circ{g}\right)}{\left(−{1}\right)}$$. Show work.
Find the composite functions $$\displaystyle{f}\circ{g}$$ and $$\displaystyle{g}\circ{f}$$. Find the domain of each composite function. Are the two composite functions equal?
$$\displaystyle{f{{\left({x}\right)}}}={x}^{{2}}−{1}$$
g(x) = −x
Given $$\displaystyle{h}{\left({x}\right)}={2}{x}+{4}$$ and $$\displaystyle{f{{\left({x}\right)}}}=\frac{{1}}{{2}}{x}+{3}$$,
$$\displaystyle{g{{\left({x}\right)}}}=\sqrt{{{5}{x}^{{2}}}}$$
$$\displaystyle{x}{\left({w}\right)}={2}{e}^{{w}}$$