The fol lowing are fundamental functions whose stated properties and graphs
you must

know.

**1 . The Constant Function**

y = f(x) = c

**Properties : **

(I) Domain :

(II) Range:

(III) y- intercept : (0, c)

x-intercept : N one except for y = f(x) = 0 (In this case the x-axis is the
graph)

(IV) Constant over
that is, always constant

(V) Symmetry: Even (y-axis symmetry)

(VI) End Behavior:

(VII) No asymptote .

**2.The Identity Function**

y = f(x) = x

**Properties :**

(I) Domain :

(II) Range:

(III) y-intercept: (0,0) ; x-intercept : (0,0)

(IV) Increasing over
,
that is, always increasing

(V) Symmetry: Odd (origin symmetry)

(VI) End Behavior :

(VII) No asymptote.

**3.The Absolute Value Function **

Properties:

(I) Domain :

(11) Range :

(III) y-intercept : (0,0) ; x-intercept : (0,0)

(IV) Decreasing over
.
Increasing over

(V) Symmetry: Even (y-axis symmetry)

(VI) End Behavior:

(VII) No asymptote.

**4. The Square Function **

y=f(x)=x^2

**Properties:**

(I) Domain :
,

(II) Range :

(III) y-intercept : (0,0) ; x-intercept : (0,0)

(IV) Decreasing over
.
Increasing over
.

(V) Symmetry: Even (y-axis symmetry)

(VI) End Behavior:

(VII) No asymptote.

**5. The Cube Function **

y = f(x) = x^3

**Properties : **

(I) Domain:

(II) Range:

(III) y-intercept : (0,0); x-intercept: (0,0)

(IV) Increasing over
;
that is, always increasing

(V) Symmetry: Odd (origin symmetry)

(VI) End Behavior:

(VII) No asymptote.

**6. The Square- Root Function **

**Properties:**

(I) Domain:

(II) Range:

(III) y-intercept: (0,0) ; x-intercept: (0,0)

(IV) Increasing over

(V) Symmetry: None

(VI) End Behavior :

(VII) No asymptote.

**7.The Reciprocal Function **

y = f(x) = 1/x

Properties:

(I) Domain :
That is, all real numbers except x = 0.

(II) Range:
That is, all real numbers except y = 0.

(I1I) y-intercept: None; x-intercept: None

(IV) Decreasing over

(V) Symmetry : Odd (origin symmetry)

(VI) End Behavior :

(approaches
0 from the left),

(approaches
0 from the right),

(VII) Vertical asymptote: x = 0 (y-axis) ; Horizontal asymptote : y = 0 (x-axis)

**8. The Exponential Function **

**Properties:**

(I) Domain:

(1I) Range:

(III) y-intercept : (0,1) ; x-intercept : None

(IV) Increasing over
;
that is, always increasing

(V) Symmetry: None

(VI) End Behavior :

(VII) Horizontal asymptote : y = 0 (the x-axis) . No vertical asymptote.

**9. The Natural Logarithm Function **

y = f(x) = ln(x)

**Properties:**

(I) Domain :

(II) Range :

(III) y-intercept: None; x-intercept : (1,0)

(IV) Increasing over
that is, always increasing

(V) Symmetry: None

(VI) End Behavior:

(VII) Vertical asymptote : x = 0 (the y-axis) . No horizontal asymptote.

**Note:**

y = e^{x} and y = ln(x) are inverse functions.

If two functions are inverses of each other then the domain of one is the range
of the

other and vice versa . For example, if (2, -3) is a point on a function, then
(-3, 2) is a point

on its inverse.

To get the graph of the inverse of a function from the graph of the function,
simply reflect

the graph about the line y = x.

So if you start out with y = e^{X} , you can get the graph of y = ln(x), simply
reflect the

graph of y = e^{x} about the line y = x .