A catalog of essential functions
Linear functions (graph is line) f(x) = mx + b
Polynomials
where n is a nonnegative integer
coefficients degree
linear when degree is 1 (graph is line)
quadratic when degree is 2 (graph is parabola)
cubic when degree is 3
Power functions
where a is a constant
a = n is a positive integer (have a polynomial in this
case)
a = 1 a = 2 a = 3 a = 4
a = 1/n, where n is a positive integer
a = 1/2 a = 1/3
a = –1 f(x) = 1/x
Rational functions
where P(x) and Q(x) are polynomials
Trigonometric Functions
Exponential functions and logarithms a is a positive constant, a ≠ 1
exponential function
domain range
logarithm
domain range
TRANSFORMATIONS of functions (for graphing)
Shifts c > 0
y = f(x) + c shifts the graph of y = f(x) up by c units
y = f(x) – c shifts the graph of y = f(x)
y = f(x – c) shifts the graph of y = f(x)
y = f(x + c) shifts the graph of y = f(x)
Stretching and Compressing c > 1
y = cf(x) stretches the graph of y = f(x) vertically by a factor of c
y = (1/c)f(x) compresses the graph of y = f(x) vertically by a factor of c
y = f(cx) compresses the graph of y = f(x) by a factor of c
y = f(x/c) stretches the graph of y = f(x) by a factor of c
Reflecting
y = –f(x) reflects the graph of y = f(x) through the
y = f(–x) reflects the graph of y = f(x) through the
EXAMPLE Graph
Combining functions
EXAMPLE For ,
find rules for each of . Also specify
the domain of each.
domain of f is domain of g is
domain | |
Composition of functions For two functions f and g,
the composition of f with g is
defined by
Domain of consists of
the numbers x that are in the domain of g AND such that g(x) is
in the domain of f.
EXAMPLE For f(x) = cos x and , find
RECALL Trig identities and unit circle