MATH 117 - Calculus 1 for Engineers
Kevin Carruthers
Fall 2012
Functions
A function is a rule which assigns a single output to a variable given at least one input
variable. In this class we deal only with a single input x and write y = f(x)
x is the indepenant variable
y is the dependant variable
A domain is the set of allowable values for the independant variable (x)
A range is the set of possible values for the dependant variable (y)
Function Composition
If y = f(x) and x = g(t) then we can view y as a function of t, y = f(g(t)), which we
sometimes write as y = f g(t)
3
For f(x) = 1 sinx;g(x) = x ,
3 3
f g(x) = f(x ) = 1 sinx
g f(x) = g(1 sinx) = (1 sinx)3
Domain of Composite Functions
For f g(x),
R = fy j y = f g(x); x Dgg
or
R = fy j y = f(x); x Rgg
1 Inverse Functions
We say g is the inverse of f if
g(f(x)) = x
for all x in the domain of f. If g(f(x)) = x, then
Rf D g
Typically, the inverse of f(x) is written as
1
g = f (x)
Note that this does not mean the reciprocal:
1 1
= (f(x))
f(x)
Finding Inverses
In simple cases, we just solve y = f(x) for x.
Example: if f(x) = x1 , nd the inverse
2
x 1
y =
2
2y = x 1
2y + 1 = x
1
f (y) = 2y + 1
1
Does f (x) always exist?
No, f(x) only has an inverse if it is one-to-one, ie. if its graph passes the horizontal line test.
Mathematically, if f(x0) = f(x1); 0 = x1
We can restrict the domain so a non-invertible function has an inverse:
p
f(x) = x on [0;1) has inverse f 1(x) = x
2 1 p
f(x) = x on (1;0] has inverse f (x) = x
2 Even and Odd Periodic Functions
A function is even if f(x) = f(x), this means its graph is symmetric about the y-axis.
A function is odd if f(x) = f(x), this means its graph is symmetric about the origin (ie.
re
ected in the x- and y-axes).
Most functions are neither even nor odd, but we can transform any function into the sum of
even and odd parts.
f(x) f(x)
f(x) = f(x) +
2 2
f(x) f(x) f(x) f(x)
= 2 + 2 + 2 2
= f(x) + f(x)+ f(x) f(x)
2 2
= g(x) + h(x)
where g(x) is an even function and h(x) is an odd function.
This idea is used to form two useful functions.
x e + ex e ex
e = 2 + 2
= coshx + sinhx
Usefulness of Symmetry
To sketch f(x) = x jxj, we note that f(x) is even, thus we can consider only x > 0 and
use symmetry to determine x < 0
Periodicity
A function f(t) is periodic with period P if
f(t + nP) = f(t); n Z
Any function which exists only over a nite interval can be extended as an even or odd
periodic function.
3 Absolute Values
(
x if x > 0
jxj =
x if x < 0
p p p
Alternatively, we can use jx . Note: x 6= x. Note 2: ( x) = x
Geometrically, jxj is the distance between x and 0 on the number line. Similarly, jx aj is
the distance between x and a on the number line.
Heavyside Function
(
0 if t < 0
H(t) =
1 if t 0
(
0 if t < a
H(t a) =
1 if t a
Multiplying a function by H(t a) is like a switch which "turns the function on" at t = a.
We can combine multiple Heavyside functions to "activate" certain functions in a specied
region.
Example: f(t) = t + (2 t)H(t + 2) + (t + 1)H(t) + (2 t t )H(t 1)
8
> t if t (1;1)
<
f(t) = 2 if t [2;0)
> t + 1 if t [0;1)
:
3 t if t [1;1)
How do we do this in reverse? ie: take a piecewise function and denite it in terms of
H(t a)?
1. Use the fact that (for a < b) 8
>
< 0 if t < a
H(t a) H(t b) 1 if a t < b
:
0 if t b
makes an on/o switch. So does (
1 H(t a) =1 if t < a
0 if t a
2. When a new function is turned on, subtract the previous one.
4
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