function - Christian Brothers University
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Transcript 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:
y2 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
xt
• 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
be
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 f ( x) a
x
• 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.