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What is a composition of functions?

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Answer: It is the application of one function to the output of the first function
In the following flow chart, our second function $$ g(x)) $$ is applied to the result of $$ f(x) $$ .

composition of functions arrow chart and flow diagram
Step By Step Flow Chart
The flow chart below shows a step by step walk through of $$ (f \cdot g)(x) $$
As you can see, you always go right to left!
Step 1) Perform right side function $$ g(x)$$
Step 2) Apply the left side function $$ f(x) $$ to the output of Step 1
flow chart step by step of composition
Try some practice probblems down below

Are compositions commutative?

Does $$ f(g(x)) = g(f ( x)) $$ ?

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What about the composition of inverse function

In other words, what if $$f(x)$$ and $$ g(x)$$ are inverses?

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Again, the best way to understand this is to try some examples, and see what happens.
Example 1
Let
$$ \bf{ \color{Red}{ f(x) = x + 3}} \\ \bf{ \color{Blue}{ g(x) = f^{-1}(x) = x -3}} $$.
What is $$ f(g(x)) = ? $$
Example 2
Let
$$ \bf{ \color{Red}{ f(x) = x^2}} \\ \bf{ \color{Blue}{ g(x) = f^{-1}(x) = \sqrt{x}}} $$.
What is $$ f(g(x)) = ? $$
question question
Practice Problems I


Directions: : Evaluate each composition of functions listed below.
Problem 1) Let $$ f(x) = x + 3$$ and $$ g(x) = 4x $$.
Evaluate $$ f( g( 2)) $$
Step 1


Problem 2) Let $$ f(x) = x^2 $$ and $$ g(x) = x+1 $$.
Evaluate $$ (f \cdot g)( 8) $$
Step 1


Problem 3) Let $$ f(x) = | 7x + 4 | $$ and $$ h(x) = \sqrt{2x}$$.
Evaluate $$ (f \cdot h)( 18) $$
Step 1
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Challenge problems Practice Problems II


Advanced problems--solving algebraically.
Problem 1) Let $$ k(x) = x^2 + x $$ and $$ h(x) = x + 3$$.
Evaluate $$ (k \cdot h)( x ) $$
Step 1




Problem 1) Let $$ g(x) = 2x^2 - 3x$$ and $$ h(x) = x + 4$$.
Evaluate $$ (g \cdot h)( x ) $$
Step 1