Building Functions From
Functions

• combining functions (+ , ─ , x , 
)
• composition of functions (incl. domain)
• decomposing functions
• graphing relations
• parametric equations
• inverses

• combining functions is easy to do, simply apply the operation to the two functions and then simplify to get a new function
• the domain of the new function consists of all numbers that belong to both domains of the original functions
• in the case of division, the zeros of the denominator are excluded from the domain

• composition of functions refers to taking one function and plugging it into the other
• the composition of f and g is denoted f g
• the domain of this composition is all of the x -values in the domain of g that map to get g ( x )-values in the domain of f
• order matters, usually
( ( ))

( ( ))

• for a given functions, h , you must find functions f and g such that: ( )

( ( ))
• example:
( )

3 x

1 then
( )

3 x
 f x
 x

• we spend most of our time studying functions, but sometimes we need to graph relations that are non-functions
• the normal mode of our calculator will not do this so we must either:
– graph it by hand by solving for y and graphing two or more “implicitly defined functions”
– graph it in the calculator using a different mode

• one way to graph non-functions in the calculator is using parametric equations
• the relation is defined by having both elements of the ordered pair (x , y) defined by a third variable, t, called the parameter
• ex. x
 
2 t y t 1 t is any real number

• to graph parametric equations in your calculator, first go to MODE and change FCN to
PAR
• go to y = , enter the two equations, notice that when the variable button is pressed, t is entered instead of an x
• go to WINDOW, make t-min = -5 , t-max = 5 and t-step = 0.1 (x and y should be normal window values, ZOOM-DEC)
• press GRAPH, use TRACE to see some points

• the ordered pair ( a , b ) is in a relation iff the ordered pair ( b , a ) is in its inverse relation
• to graph an inverse relation, simply take several of the points of the original relation, flip their coordinates, and graph the new points
• sometimes the inverse of a function is also a function
(the original must pass a horizontal line test); these functions are called one-to-one
• for one-to-one functions, the domain of the original becomes the range of the inverse and vice versa

Inverses
• to find the inverse of a relation algebraically, switch the x
’s and y
’s and then solve for y
(state any restrictions on the domain if necessary)
• the graph of a relation and its inverse will be reflections of each other across the line y = x
• if f and g are inverse functions then
 x

( ( ))