A Library of Functions
Definition of a Function
1. Definition: y is a function of x if we can associate a unique value of y
to each value
of x. We call y the dependent variable and x the independent variable where
y = f(x).
2. The domain of a function is the set of all possible values of the independent variable.
The range of the function is the corresponding set of values of the dependent variable.
3. Proportionality :
y is directly proportional to x if there exists a constant k such that
y = kx.
y is inversely proportional to x if there exists a constant k such that
y = k/x.
(i.e. y is proportional to the reciprocal of x).
1. Definition: A function is linear if any change in the independent variable
proportional change in the dependent variable.
2. Representation of a linear function:
y = f(x) = mx + b
where m is the slope of the graph and b is the value of y when x = 0.
To compute the slope:
We can also describe m as the change in y for a unit change in x.
3. Given a point (x1, y1) on a line and the slope m of the line, we can write
equation of the line as
y − y1 = m(x − x1).
4. A function is linear if and only if for equal differences in x values
there is a constant
difference in y values. How does this relate to the slope of the function?
1. Definition: A function is quadratic if it satisfies the formula y = f(x) = ax2+bx+c.
2. The shape of the graph is determined by whether a > 0 (shaped like x2) or
a < 0
(shaped like −x2).
3. The roots of the function are at
In particular the quadratic has no roots if
one root if
and two roots if
4. A quadratic function has a minimum value at when a > 0.
5. A quadratic function has a maximum value at when a < 0.
Polynomial and Rational Functions
1. A power function is a function where the dependent variable is
proportional to a
power of an independent variable. The general form is:
where k and p are constants.
2. A polynomial is a function of the form
where each of is a constant.
The largest power of x for which the coefficient of x is non zero is called
of the polynomial.
3. The graph of an nth degree polynomial turns around at most n − 1 times. An
degree polynomial will have at most n roots or zeros.
the polynomial p(x) will look like the power function
is the degree of the polynomial.
5. A rational function is a function of the form
where p(x) and q(x) are polynomials.
6. r(x) will have vertical asymptotes whenever q(x) = 0 (after simplification) .
Inverse of a function
1. A function y = f(x) will have an inverse if and only if for every y there
is a unique
x that maps to it.
2. Computing inverses from a table- just read the table backwards -i.e. your
variable is the right column and your dependent variable is the left column.
3. To compute inverses algebraically, given y = f(x), solve for x in terms of y.
4. Given y = f(x) an invertible function, the domain of f-1(x) is the range
of f and
the range of f-1 is the domain of f.
5. To graph the function f-1(x) given the graph of f(x) reflect the graph of
the diagonal y = x.
6. A function will have an inverse if and only if the graph of the function
every horizontal line at most once.
7. A function that is only increasing or only decreasing will have an inverse.
8. Sometimes a function might have an inverse in a specific domain. For
f(x) = x2
has an inverse if the domain of the function is restricted to only the positive numbers.
9. If a function has a horizontal asymptote at y = k then the inverse
have a vertical asymptote at x = k.
Transforming and combining functions
1. Shifts and stretches: Given a function y = f(x),
(a) Replacing y by y − k moves a graph up by k or down if k is negative. Ie.,
k is positive, y = f(x) + k has the same graph as y = f(x) but shifted up on
the y axis by k ( and down if k is negative.)
(b) Replacing x by x − k shifts a graph to the right by k if k is positive
the left if k is negative. Ie., y = f(x−k) has the same graph as y = f(x) but
shifted to the right k units if k is positive and shifted to the left k units if k
(c) Multiplying the function by a constant k:
If k > 0, the function y = kf(x) will look like y = f(x) except stretched
vertically if k > 1 or shrunk if 0 < k < 1.
If k < 0, y = kf(x) will look like y = f(x) reflected about the x-axis and then
stretched if k < −1 and shrunk otherwise.
2. Sums of Functions :
If we are given two (or 3, 4,...) functions f(x) and g(x) then the sum of the two
functions is also a function. That is, we can create a new function h(x) where the
value of h(x) at any point the sum of the values of f(x) and g(x) at that point.
Thus we can write
h(x) = f(x) + g(x).
The domain of h will be all x where both f(x) and g(x) are defined. Similarly
difference of two functions (f(x) − g(x)) will also be a function.
3. Composition of functions:
A composite function or a function of a function is a function C(x) which can be
written as C(x) = f(g(x)) for two functions f(x) and g(x).
Given an x we can compute C(x) by first computing y = g(x) and then computing
z = f(y). Thus C(x) = z.
4. The domain of C(x) is all x such that g(x) exists and g(x) is in the domain of f.
5. In general f(g(x)) ≠ g(f(x)).
6. The composition of a function f(x) and its inverse f-1(x):
f(f-1(x)) = f-1(f(x)) = x.
1. Definition of an exponential function:
where P0 is the initial quantity (when t = 0) and a is the base of the function).
2. A function is exponential if and only if for equal differences in t
values, there is a
constant ratio in P(t) values. This ratio is the base a of the function.
3. Restrict a such that 0 < a < 1 and a > 1.
Exponential Growth: a > 1
Exponential Decay: 0 < a < 1
4. Doubling time of a function with exponential growth is the time taken (ie.,
in t) for the function value to double.
We can calculate the doubling time by solving for t in the equation at = 2.
If P = , then the doubling time is
5. Half-life of a function that is decaying exponentially is the time taken (ie.,
t) for the value of the function (dependent variable to become half the original value.
We can calculate the half-life by solving for t in the equation at = 1/2.
If , then the half-life is
6. Alternative formula for exponential functions: Can write a as 1 + r where
r is the
growth rate of the function and a as 1−r where r is the decay rate of the function.
Thus the corresponding functions will be
7. Rules for computing Exponents:
1. f(x) = loga(x) is the inverse function of y = ax. logax = c if ac = x..
2. logax is not defined if x is negative or if x = 0.
3. Rules for Computing logs:
4. Logs just simplify the level of difficulty - Logs convert exponentiation
and multiplication to addition.
The Number e and Natural Logarithms
1. ln x = logex = c if ec = x.
2. The rules for computing ln x are the same as the rules for computing log x.
3. ex and ln x are inverse functions. Ie, eln x = ln ex = x.
4. We can write any exponential growth function in the form
and any exponential decay function where k is positive.
5. In the above equation, we need to know how to convert between at and ekt. Substituting
above we see that a = ek or k = ln a.
6. Can use either ln or log to solve for x in .
Properties of Functions
1. Increasing and Decreasing Functions:
A function is increasing if as x increases, y = f(x) increases.
A function is decreasing if y = f(x) decreases as x increases.
2. Odd and Even Functions:
A function f is an even function if f(x) = f(−x) for all x. That is, f is even if the
graph of f is symmetric about the y-axis.
A function f is an odd function if f(−x) = −f(x) for all x, That is, f is odd if the
graph of f is symmetric about the origin.
3. A graph is concave up if
(a) if the function is increasing and the increase is
That is, in the table of x and y values, for equal difference in x values, the
difference in y values is increasing as x is increasing.
(b) if the function is decreasing and the decrease is
That is, for equal differences in x values, the difference in y values is decreasing
(as x is increasing).
4. A graph is concave down if
(a) if the function is increasing and the increase is decreasing.
That is, in the table of x and y values, for equal
difference in x values, the
difference in y values is decreasing as x is increasing.
(b) if the function is decreasing and the decrease is increasing.
That is, for equal differences in x values, the difference
in y values is increasing
(as x is increasing).