1. Building Functions From Functions
Section 1– 4
Building Functions From Functions

2. Section 1-4 combining functions (+ , ─ , x ,  )
composition of functions (incl. domain)
decomposing functions
graphing relations
parametric equations
inverses

3. Combining Functions
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

4. Composition of Functions
composition of functions refers to taking one function and plugging it into the other
the composition of f and g is denoted
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

5. Decomposing Functions
for a given functions, h, you must find functions f and g such that:
example:

6. Graphing Relations
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

7. Parametric Equations
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.

8. Parametric Equations
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

9. Inverses
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

10. 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