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5

BASIC GRAPHS

A constant function


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

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

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

Parabola and square root functions

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

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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