y = (formula that contains only x variable). We say that a function is given implicitly, if it is
given by an equation that is not in the above form. For example, y = 3x + 2 is a function given
in an explicit form, and 3x - y + 2 = 0 is the same function given in an implicit form.
When we use a formula to define a function we often give a name to a function(f,g,h,etc.) and
use a special notation for the output, y. If x is the input, then we denote the output by f(x) (read
“f of x”).
Caution: f(x) is not a multiplication of f and x. It is an entity that can’t be split. f(x) denotes
output that corresponds to the input x.
For example, instead of writing y = 3x+2, we often write f(x) = 3x+2. With this notation f(1)
denotes the output (y) that correspond to the input 1, that is f(1) = 3∙(1)+2= 5.
If we say that f(4) = -5, that means that when the input is 4, the output is -5. This also means
that the function f contains a pair (4, -5)
When a function f is given by a formula and its domain is not given, then it is assumed that the
domain is the largest set of real numbers for which f (x) can be computed and is a real number.
An equation defines a function if it can be solved for y and the solution is unique.
Example: 3x+2y = 4 is a function, because we can solve it for y, y = (-3/2)x + 2 and the
solution is unique. The equation x2
+ y2
= 1 is not a function, because when we solve it for y,
we get
2
y 1 x
, two solutions.
(iv) The graph, which is the set of all pairs (x,y) that are plotted in the coordinate system.
The graph is the set of all points (x, f(x)), where x belongs to the domain of a function f.
A point (a,b) is on the graph of a function f, if and only if b = f(a)
A graph represents a function, if it passes the Vertical Line Test, which says that If any
vertical line crosses the graph at most at one point, then the graph is that of a function.
The domain of a function given by a graph is the set of all x -coordinates of the points on the
graph. The range is the set of all y-coordinates of the points on the graph
Example: Find f(0), f(1), f(-2), f(-x), -f(x), f(x+1), f(2x), f(x+h) for f(x) = 3x2
-2x – 4
f(0) = 3(0)
2
-2(0) – 4 = -4
f(1) = 3(1)
2
– 2(1) – 4 = -3
f(-2) = 3(-2)
2
-2(-2)- 4= 12
f(-x) = 3(-x)
2
-2(-x) -4 = 3x2
+ 2x – 4
f(x+1) = 3(x+1)
2
-2(x+1) – 4 = 3(x2
+ 2x +1)-2x-2 -4 = 3x2
+ 6x + 3 -2x – 6= 3x2
+ 4x – 3
f(2x) = 3(2x)
2
-2(2x) – 4 = 3(4x2
) – 4x – 4= 12x2
– 4x – 4
f(x+h) = 3(x+h)
2
-2(x+h) – 4 = 3(x2
+ 2xh +h2
) – 2x- 2h -4 = 3x2
+ 6xh + 3h2
-2x – 2h – 4