# Composite functions

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##### Intros
###### Lessons
1. What is a "Composite Function"?
2. Quick review on basic operations with functions.
3. Composition of Functions: putting one function inside another function!
##### Examples
###### Lessons
1. Introduction to Composite Functions
If $f(x)= 4x-5$
$g(x)=8x^2+6$
Determine
1. $(f\circ g)(x)$
2. $(g\circ f)(x)$
3. $(f\circ f)(x)$
4. $(g\circ g)(x)$
5. $(f\circ g)(2)$
evaluate in two different ways
2. Evaluate a Composite Function Graphically
Use the graphs of $f(x)$ and $g(x)$ to evaluate the following:

1. $f(g(-4))$
2. $f(g(0))$
3. $g(f(-2))$
4. $g(f(-3))$
3. Determine the Composition of Three Functions
Use the functions $f(x)=3x,$
$g(x)=x-7$
and $h(x)=x^2$
to determine each of the following:
1. $(f\circ g\circ h)(x)$
2. $g(f(h(x)))$
3. $f(h(g(x)))$
4. $(h\circ g\circ f)(x)$
4. Determine the Original Functions From a Composite Function
If $h(x)=(f\circ g)(x)$
determine $f(x)$
and $g(x)$
1. $h(x)=(7x-5)^3-4(7x-5)+1$
2. $h(x)=\sqrt{4x^3-9}$, give two possible sets of solutions
5. Composite Functions with Restrictions
Consider $f(x)=\sqrt{x-13}$
and $g(x)=x^2+4$, for each of the function below:
1. Determine:
i)
$(f\circ g)(x)$
ii)
$(g\circ f)(x)$
2. State the domain and range of:
i)
$f(x)$
ii)
$g(x)$
iii) $(f\circ g)(x)$
iv)
$(g\circ f)(x)$
3. Sketch the graph of:
i)
$f(x)$
ii)
$g(x)$
iii) $(f\circ g)(x)$
iv)
$(g\circ f)(x)$
6. Composite Functions with Restrictions
Consider $f(x)=\sqrt{x-13}$
and $g(x)=x^2+4$, for each of the function below:
i) state the domain and range
ii) sketch the graph.
1. $f(x)$
2. $g(x)$
3. $(f\circ g)(x)$