Functions, Limits, and Continuity

A function is a rule that assigns to each element x from a set known as the "domain" a single element y from a set known as the "range". For example, the function y = x² + 2 assigns the value y = 3 to x = 1, y = 6 to x = 2, and y = 11 to x = 3. Using this function, we can generate a set of ordered pairs of (x, y) including (1, 3),(2, 6), and (3, 11). We can also represent this function graphically, as shown below.

Note that in the graph above, each element x is assigned a single value y. If a rule assigned more than one value y to a single element x, that rule could not be considered a function. As you may recall from precalc, we can test for this property using the vertical line test, where we see whether we can draw a vertical line that passes through more than one point on the graph:

Because any vertical line would pass through only one point, y = x² + 2 must be assigning only one y value to each x value, and it therefore passes the vertical line test. Thus, y = x² + 2 can rightfully be considered a function.

Although a function can only assign one y value to each element x, it is allowed to assign more than one x value to each y. This is the case with our function y = x² + 2. The value x = 4 is mapped to the single value y = 18, but the value y = 18 is mapped to both x = 4 and x = - 4.

A one-to-one function is a special type of function that maps a unique x value to each element y. So, each element x maps to one and only one element y, and each element y maps to one and only one element x. An example of this is the function x³:

We can see that it is a function because it passes the vertical line test. We can also see that it assigns only one x value to each y value. Thus, it is a one-to-one function. Again from precalculus, we can see graphically whether a function is a one-to-one function by using the horizontal line test:

Any horizontal line we draw through the graph of the function y = x³ passes through only one point, so it must be assigning only one x value to each y, and can therefore be considered a one-to-one function. Horizontal lines through y = x² + 2 pass through more than one point, so this function fails the horizontal line test.

In summary, for a rule to be a function, its graph must pass the vertical line test. To be a one-to-one function, it must pass both the vertical line test and the horizontal line test.

In this guide, we will often give functions names such as f (x), g(x), h(x), etc. For example, when we say "f (x) = x² + 2", we mean for f (x) to refer to the rule that assigns the number y = x² + 2 to any real number x.

As we proceed, two types of functions to be aware of are polynomial functions and rational functions.

A polynomial function is any function of the form

where a₀, a₁, a₂,...a_n are constants and n is a nonnegative integer. n denotes the "degree" of the polynomial.

You should be familiar with the common names of certain polynomial functions. A second-degree polynomial function is a quadratic function (f (x) = ax² + bx + c). A first-degree polynomial function is a linear function (f (x) = ax + b). Finally, a zero-degree polynomial function is a simply a constant function (f (x) = c).

A rational function is a function r of the form

where f (x) and g(x) are both polynomial functions. For example,

is a rational function. Note that we must exclude from the domain of r(x) any value of x that would make the denominator, g(x) equal zero, since this would make r(x) undefined. Thus, x = 0 is not in the domain of the function r(x) we just defined above.

Another useful classification of functions is even and odd. For an even function, f (- x) = f (x) for all x in the domain. This sort of function is symmetric with respect to the y-axis. For example:

For an odd function, f (- x) = - f (x) for all x in the domain. This sort of function is symmetric with respect to the origin. For example:

As we know, f is a function that can take an input x and transform it into an output f (x). Similarly, f can take the output of another function, such as g(x) as its input, and transform that input into f (g(x)). When two functions are combined so that the output of one function becomes the input for the other, the resulting combined function is called a composite function. The notation for the composite function f (g(x)) is (fog)(x).

Example:

If f (x) = 3x + 4 and g(x) = 2x - 7, then how could we find (fog)(2)?

Solution:

The problem is asking us to find f (g(2)). One way is to work step-by-step with g and then with f:

g(2)
= 2(2) - 7
= -3


Now we use g(2) = - 3 as the input for f:

f (g(2))
= f (- 3)
= 3(- 3) + 4
= -5

A second way would be to solve for (fog)(x) directly.

f (g(x))
= f (2x - 7)
= 3(2x - 7) + 4
= 6x - 21 + 4
= 6x - 17

Now, we can plug x = 2 into this function: f (g(2)) = 6(2) - 17 = - 5

One type of function we'll be dealing with often in calculus is the piecewise-defined function. These functions are defined differently for different intervals in their domain. For example, consider the following piecewise function:

For x less than or equal to 2, f (x) is defined by f (x) = x². For x greater than 2, f (x) is defined by f (x) = 2x. Thus, f (1) = 1² = 1, and f (4) = 2(4) = 8. The graph of this function is below:

Finally, we should briefly mention interval notation, which we'll be using throughout the rest of the guide. An interval is a set of all numbers between two endpoints. An closed interval includes both of the endpoints, while an open interval includes neither of the endpoints. So, [a, b] means the set of all x such that a≤x≤b (closed interval) (a, b) means the set of all x such that a < x < b (open interval) Intervals can also be half-open (and half-closed). For example, [a, b) is closed at x = a and open at x = b. This interval represents a≤x < b Intervals that have infinity as an endpoint should always be open at infinity, since no interval can actually contain infinity. Thus, "all numbers less than 4" should be written as (- ∞, 4] , while "the set of all real numbers" should be written as (- ∞,∞).