Download function - Christian Brothers University

Calculus I
Hughes-Hallett
Math 131
Br. Joel Baumeyer
Christian Brothers University
Function: (Data Point of
View)



One quantity H, is a function of
another, t, if each value of t has a
unique value of H associated with it.
In symbols: H = f(t).
We say H is the value of the function
or the dependent variable or output;
and
t is the argument or independent
variable or input.
Working Definition of
Function: H = f(t)
A function is a rule (equation) which
assigns to each element of the domain
(independent variable) one and only one
element of the range (dependent variable).
Working definition of function
continued:
Domain is the set of all possible values of
the independent variable (t).
Range is the corresponding set of values of
the dependent variable (H).
Questions?
General Types of
Functions (Examples):





Linear: y = m(x) + b; proportion: y = kx
Polynomial: Quadratic: y =x2 ; Cubic: y= x3 ;
etc
Power Functions: y = kxp
Trigonometric: y = sin x, y = Arctan x
Exponential: y = aebx ; Logarithmic: y = ln x
Graph of a Function:
The graph of a function is all the points in
the Cartesian plane whose coordinates make
the rule (equation) of the function a true
statement.
Slope
• m - slope :
y2  y1 y rise
m


x2  x1 x run
b: y-intercept
• a: x-intercept
• . x , y  and  x , y  are po int s
1
1
2
2
5 Forms of the Linear Equation
• Slope-intercept: y = f(x) = b + mx
• Slope-point: y  y1  m ( x  x1 )
• Two point:
y 2  y1
y  y1 
( x  x1 )
x2  x1
• Two intercept: x  y  1
a
b
• General Form: Ax + By = C
Exponential Functions: P  P0 a
If a > 1, growth;
a<1, decay
• If r is the growth rate then a = 1 + r, and
P  P0 a  P0 (1  r )
t
t
P0
• If r is the decay rate then a = 1 - r, and
P  P0 a  P0 (1  r )
t
t
t
Definitions and Rules of
Exponentiation:
• D1:a 0  1, a 1  1a , and a  x  1x , a  0
a
1
1
• D2:a 2  a and a n  n a ; a  0 for n even
• R1: a x a t  a x  t
x
a
xt
• R2:
 a
t
a
• R3:
a   a
x t
xt
1
Inverse Functions: g ( x)  f ( x)
• Two functions z = f(x) and z = g(x) are
inverse functions if the following four
statements are true:
• Domain of f equals the range of g.
• Range of f equals the domain of g.
• f(g(x)) = x for all x in the domain of g.
• g(f(y)) = y for all y in the domain of f.
A logarithm is an exponent.
.
log10 x  c means 10  x
c
ln x  log e x  c means e  x,
c
and in general :
log a b  c means a  b
c
General Rules of Logarithms:
log(a•b) = log(a) + log(b)
log(a/b) = log(a) - log(b)
log( a )  p log( a )
p
log c c  x and c
x
log c x
x
log c 1  0 because c  1, c  0
0
log c a
also log b a 
log c b
e = 2.718281828459045...
• Any exponential function
y  ab
kx
can be written in terms of e by using the fact
ln b
that
be
So that y  ab kx becomes y  a ( e ln b ) kx
Making New Functions from Old
Given y = f(x):
(y - b) =k f(x - a) stretches f(x) if |k| > 1
shrinks f(x) if |k| < 1
reverses y values if k is negative
a moves graph right or left, a + or a b moves graph up or down, b + or b If f(-x) = f(x) then f is an “even” function.
If f(-x) = -f(x) then f is an “odd” function.
n
Polynomials:
y   ak x k  a0 x 0  a1 x    an x n
k 0
• A polynomial of the nth degree has n roots
if complex numbers a allowed.
• Zeros of the function are roots of the
equation.
• The graph can have at most n - 1 bends.
• The leading coefficient an determines the
position of the graph for |x| very large.
Rational Function: y = f(x) = p(x)/q(x)
where p(x) and q(x) are polynomials.
• Any value of x that makes q(x) = 0 is called a
vertical asymptote of f(x).
• If f(x) approaches a finite value a as x gets larger
and larger in absolute value without stopping, then
a is horizontal asymptote of f(x) and we write:
lim
x
f ( x)  a
• An asymptote is a “line” that a curve approaches
but never reaches.
Asymptote Tests y = h(x) =f(x)/g(x)
• Vertical Asymptotes: Solve: g(x) = 0
If y    as x  K, where g(K) = 0,
then x = K is a vertical asymptote.
• Horizontal Asymptotes:
If f(x)  L as x   then y = L is a vertical
asymptote. Write h(x) as:
h(x) 
f (x)
g( x )
1
, where n is the highest power of x in f(x) or g(x).
xn
1
xn
Basic Trig
• radian measure: q = s/r and thus s = r q,
• Know triangle and circle definitions of the
trig functions.
• y = A sin (Bx + C) + k
• A amplitude;
• B - period factor; period, p = 2p/B
 j - phase shift; j  -C/B
• k (raise or lower graph factor)
Continuity of y = f(x)
• A function is said to be continuous if there
are no “breaks” in its graph.
• A function is continuous at a point x = a if
the value of f(x)  L, a number, as x  a
for values of x either greater or less than a.
Intermediate Value Theorem
• Suppose f is continuous on a closed interval
[a,b]. If k is any number between f(a) and
f(b) then there is at least one number c in
[a,b] such that f(x) = k.