Math 104 Section 101 Notes for 2013/9/5
September 5, 2013
Calculus is the study of functions on real numbers. In this course we will
look primarily at functions used in business and economics: for example, as a
company varies the number of widgets q that it produces, the price p that it
can sell them at will change, producing a relationship between p and q.
De▯nition 1. a function y = f(x) is a rule that applies to some set of real
numbers D, and for each x in the set D, it assigns it a unique real number y.
The set D is called the domain of f.
Example 2. If f(x) = x , then we can take the domain D to be the set of all
Example 3. If f(x) = 1=x, then f is de▯ned at every point x 6= 0.
Example 4. If f(x) = x, then f is de▯ned at every x ▯ 0.
We can de▯ne a function by its graph, and much of calculus comes from draw-
ing connections between the algebraic side (i.e. an equation) and the graphical
side (i.e. the graph of a function). For example, here is the graph of f(x) = x :
Using the graph, we can determine whether a curve is a function or not.
Because a function must assign to each x in the domain a unique value y, each
vertical line intersects the graph at only one point (x;y). We can graphically
see that the curve of x indeed represents a function. In contrast, the curve
below does not represent a function:
1 Note that the dashed vertical line intersects the graph at multiple points,
so it cannot assign a unique y to each x. This is called the vertical line test:
a graph represents a function if and only if each vertical line intersects it at at
most one point.
Given the importance of vertical lines in determining whether a graph is a
function, it is natural to ask what horizontal lines mean. They have a meaning,
but ▯rst, we need a de▯nition:
De▯nition 5. A function f is called 1-to-1 if for each y there is at most one
x such that y = f(x). Equivalently, we call f invertible.
The relationship between x and y can sometimes be switched. In economics,
a demand curve relates price and quantity; price can be viewed as a function of
quantity produced, but the quantity demanded can also be viewed as a function
of the price charged. To see if we can replace the relationship y = f(x) with
x = f (y), we need to check that f would be a function. For that, we require
f to be 1-to-1, and this is the case if and only if f satis▯es the horizontal line
test: the graph of a function represents a 1-to-1 function if and only if each
horizontal line intersects it at at most one point.
Example 6. The following function is not 1-to-1, as demonstrated by the hori-
zontal line depicted:
2 Example 7. The inverse of a powerpfunction is a root function. The function
y = x is 1-to-1 and its inverse is3x, and more in general if a is an odd integer
then y = x is invertible and its inverse isax.
Remark 8. As can be seen in the example above, the graph of the inverse of a
function is ob