Composite Functions

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Composite Functions

• Functions must have 1 element from the domain that assigns to one and only one element from the range
• i.e. one x value is associated with only one y value
• A composite function is when one function is processed by another. The notation for composite functions looks like the following

$\text{The composition of g with f is }g(f(x))\text{ or }g\circ f$

• For $g(f(x))$ to exist, the range of $f(x)$ must be within the domain of $g(x)$
  • i.e. for $x$ in the domain of $g(x)$, $f(x)$ must exist

Inverse Functions

If we have a function, $f(x)$ that solves for values of $y$, it is possible for an inverse function, $f^{-1}(x)$, to exist such that the inverse function solves for values of $x$ in the original function $f(x)$. However, there are requirements that need to be met for an inverse function to exist

1. One to one (for both $f(x)$ and $f^{-1}(x)$): One element from the domain outputs one element in the range
2. $f^{-}1(x)$ exists naturally for a one to one function $f(x)$

The inverse of a function is its reflection over the line $y=x$ The domain of x is now the range of y The range of y is now the domain of x

Where it doesn’t exist

Many to one: More than one element from the domain outputs one element in the range. One to many: Thus, for the inverse of $f(x)$, for every $x$ value, $f^{-1}(x)$ outputs more than one element in the range

How to find the inverse of a function

$\text{We have a function }f(x)=2x+5$ $\text{If }y=f(x)$ $\text{The inverse is }x=2y+5$ $\frac{x-5}{2}=y=f^{-1}(x)$

1. Swap $y$ and $x$
2. Isolate $y$
3. Done!!!!!

Absolute Functions

• for a functions $y=|x|$, all y values are positive

To graph $y=|f(x)|$

1. Retain the part of the graph above the x-axis
2. Reflect the part of the graph with negative y values above the x-axis

To graph $y=f(|x|)$

• Retain the part on the RHS of the y axis
• Reflect these points over the y-axis, but keep the RHS points

Solving Absolute Functions

Say we have an equation $|x-a|+|x-b|$ How would we solve this?

Rational Functions

• Horizontal asymptote is

$\frac{f(x)}{g(x)}$ 2