Practice Exercises for Composite Functions

Given $f(x)=5x-5$ and $g(x)=5x-1,$
Evaluate the composite function $g[f(0)]$

The number of electric scooters e that a factory can produce per day is a function of the number of hours h it operates and is given by ${\displaystyle e\left(h\right)=290h,0\le h\le 10.}$
The daily cost c to manufacture e electric scooters is given by the function ${\displaystyle c\left(e\right)=0.05e^2+65e+1000.}$
(a) Find ${\displaystyle (c\cdot e)\left(h\right).}$
(b) Evaluate ${\displaystyle (c\cdot e)\left(13\right).}$

The given pair of functions should be rewritten as one composite form
${\displaystyle g\left(x\right)=\sqrt{5x^2}}$
${\displaystyle x\left(w\right)=2e^w}$
$g(x(ww))=?$
Evalute the composite function at 1.
$g(x(1))=?$

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
$f(x)=3x+1$
$g(x)=-x$

For $f(x)=6/x$ and $g(x)=6/x$, find the following functions.

a) $(f\cdot g)(x)$

b) $(g\cdot f)(x)$

c) $(f\cdot g)(7)$

d) $(g\cdot f)(7)$

Suppose that the dunctions q and r are defined as follows.
${\displaystyle q\left(x\right)=x^2+7}$
$r(x)=\sqrt{x+8}$
Find the following.
${\displaystyle q\cdot r\left(1\right)=?}$
${\displaystyle (r\cdot q)\left(1\right)=?}$

Let f(x) = ${\displaystyle 4x^2–6}$ and ${\displaystyle g\left(x\right)=x–2.}$
(a) Find the composite function ${\displaystyle (f\circ g)\left(x\right)}$ and simplify. Show work.
(b) Find ${\displaystyle (f\circ g)(-1)}$. Show work.