5
BASIC GRAPHS
THE FOLLOWING ARE THE GRAPHS that occur throughout analytic geometry and calculus. The student should be able to recognize them -- and sketch them -- purely from their shape. It is not necessary to plot points.
A constant function
Here is the graph of y = f(x) = 3. It is a straight line parallel to the x-axis. It is called a constant function, because to every value of x there corresponds the same value of y: 3.
Is a constant function single-valued? Yes, it is, because to each value of x there is one and only one value of y. 3.
A constant function has the form
y = c ,
where c is a constant, that is, a number.
The identity function and the absolute value function
y = x is called the identity function, because the value of y is identical with that of x. The coördinate pairs are (x, x).
In the absolute value function, the negative values of y in the identity function are reflected into the positive side. For, |−x| = |x| = x. The coördinate pairs are (x, |x|).
Example.
a) What is the domain of the identity function?
There is no natural restriction on the values of x. Therefore, the domain -- where the function "lives" -- includes every real number.
−
< x <
Note first that infinity "" is not a number nor is it a place. It is a manner of speaking. We mean, "x could take any value however far to the right on the number line (x < ), or however far to the left
(− < x)."
Note also that we write "x less than ." Equal to infinity makes no sense.
b) What is the range of the identity function?
The range are those values of y that correspond to the values in the domain. Inspecting the graph will show that y, also, will take every real value.
− < y <
Parabola and square root function
In the parabola y = x², the coördinate pairs are (x, x²). We can see that the following points are on the graph: (1, 1), (−1, 1), (2, 4), (−2, 4), and so on.
The graph of the square root function is related to y = x². It is its inverse. The coördinate pairs are (x, 
). For example, (1, 1), (4, 2), (9, 3), and so on.
Notice that the square root function is defined only for non-negative values of x. For, the square root of a negative number is not real.
Problem 1. What is the domain of the function y =x², and what is its range?
This function is defined for all values of x : −∞ < x < ∞.
As for the range, the lowest value of y is 0. And there is no limit to the highest value. 0 ≤ y ∞.
Problem 2. What is the domain of the square root function, and what is its range?
As we have stated, the square root function is defined only for non-negative values of x. Domain : x ≥ 0.
As for the range, the lowest value of y is 0. And there is no limit to the highest value. 0 ≤ y < ∞.
(If you are not viewing this page with Internet Explorer 6, then your browser may not be able to display the symbol ≤, "is less than or equal to;" or ≥, "is greater than or equal to.")
The cubic function
The cubic function is y = x³. When x is negative, y is negative -- odd powers of a negative number are negative.
Problem 3. What is the domain of the cubic function, and what is its range?
Domain: −∞ < x < ∞.
Range: −∞ < y < ∞.
Next Topic: The vocabulary of polynomial functions
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