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Functions, Limits, Continuity


Brief Review of Functions

Summary Brief Review of Functions

f (x) = a0 + a1x + a2x2 + + anxn    

where a0, a1, a2, 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).

Rational Functions

A rational function is a function r of the form

r(x) =    

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

r(x) =    

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.

Even and Odd Functions

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:

Figure %: Even functions, such as f (x) = x2 + 3, are symmetric with respect to the y-axis

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:

Figure %: Odd functions, such as f (x) = x3, are symmetric with respect to the origin

Composite Functions

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).


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


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

= 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

Piecewise-Defined Functions

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:

f (x) =    

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:

Figure %: Graph of the piecewise-defined function above

Interval Notation

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 axb (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 ax < 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 (- ∞,∞).