FUNCTIONS There are numerous cases where a second quantity is found from a first quantity by applying some rule. a) If one drops a stone from the top of a tower, how far it has fallen at a given moment (let that be h feet) can be found from the time of falling ( t seconds) by using the rule h = 16t 2 . b) If the temperature in Fahrenheit scale is F and we want to find the equivalent 5f f f temperature in Celsius scale (say C) we can use the rule C = F @ 32 . 9 c) The area A of a circle can be found from its radius r by using the rule A = π r 2
`
a
In case (a) distance is a function of time, in case ( b) Celcius temperature is a function of Fahrenheit temperature, in case ( c) area of a circle is a function of its radius. A function is a Rule. Usually we use small letters f, g, h ...... to denote different functions or different rules, 2 such as f x = x 2 + 1, g x = 10x , h x = X @ 3 x + 2 etc .
`a
`a
`a
Example : In case (a) above, for dropping the stone from the tower, we could have written h = f t = 16 t 2 where we find h by applying the rule f on t . So, “f” is the rule “ 16 B squa square re of time time in seco second ndss “.
`a `a `a `a
Her Heree f 1
=
16 B1
2
means the stone stone drops drops 16 feet feet in 1 second, = 16 means
f 3
=
16 B 3
2
means the stone stone drops drops 144 feet feet in = 144 means
3 seconds,
f 5
=
16 B 5
2
means the stone stone drops drops 400 feet feet in = 400 means
5 seconds A
`a
Usually a function f(x) is also written as y i.e. f(x) = y. So, y = x 2 + 1 and f x x. are the same function. We say, y or f(x) is a function of x
2
= x +
1
Definition : Let A and B be two non-empty sets. A function A function f from A to B is a rule that associates, with each value of x in set A, exactly one value f(x) in set B.
The function is indicated by the notation f : A Q B . f(x) is read “ f of x”. x”. We usually B are sets of real numbers. consider functions for which sets A and B The set A, which contains all possible values of x is called the Domain , and the set B which contains the corresponding values of f(x) or y is called the Range of the function (Range consists only of those elements of B which are actually paired with elements of A).
We call x the independent variable because we choose it first, and from that we calculate y, which is called dependent variable. If the function f(x) is given as an algebraic expression of x and the domain is not stated, then we take the domain as the set of all real numbers x for which f(x) is real.
The parentheses in f ( x x) (" f of x x") do not mean multiplication. They are part of what is called functional notation. f is the name of the function. And x, which is placed within the parentheses, is called the argument of the function. It is upon the argument that the function(rule) called f will "operate." Thus, the function f has been defined as follows: f ( x x) = x² + 1.
This means that f will square its argument, and then add 1. x, we get only one value of y (say when x=4, y=16 ). In y = x 2 , for any value of x, ). Here y x. is a function of x 2 But in y = x when we put x=4, we get y = + 2 or y = @ 2 i.e. we are getting more than y. So, here y is not a function of x x. one value of y
`a b c `a ` a
Example : For the function f x
= x
2
w w w w w w
a) Evaluate f 2 , f @ 5 and f p 3 b) Find the domain and range of f .
Solution : x in f(x) we get a) Substituting the values of x
`a
f 2
2
` a` a
2
f @ 5
=2 = 4
=
@
5
=
25
w w w w w w f p 3
w w w w w w2 p 3 =3
b cb c =
b) The domain is R, the set of all real numbers as any real number can be put in x here and the corresponding f(x) is a real number. place of x 2 x, f x ≥ 0. So Range of f f is y | y ≥ 0 or the Since x ≥ 0 for all real values of x
B c
interval 0 , 1
A
`a
R
S
`a
Example : Let f x
= 2 x
2
@
5 x
+
` a f ffgf bp wwwwwwwc `fffffffffffffaffffffffff`fffffafffff
3 . Evaluate f 1 , f
@
1 ,f 2
a ,
f 2 + h @ f 2 h
Solution : Substituting the values of x in f(x) we get 2 f 1 = 2 A 1 @ 5.1 + 3 = 2 @ 5 + 3 = 0
`a f g f g f g b c b c b c C ` a `a B` a ` a C B Db c ` a E@ A B C@ A
1f f f= 2 A f @ 2
2
1f f f @ 2
@
5
@
1f f f+ 3 2
w w w w w w w w w w w w w w2 w w w w w w w p p f a = 2 a @ 5 p a + 3
=
=
1f f f+ 5f f f + 3=6 2 2
w w w w w w w
2a @ 5 p a
+
3
2
2
2 2 + h @ 5 2 + h + 3 @ 2 A 2 @ 5.2 + 3 @ f f hf f 2f 2f ff f+f ff ff ff ff ff ff ff ff f f ff ff ff f f f f f f ff f f f ff f f f f ff f f f ff f f f f ff f f f ff f f f ff f f f f ff f f f ff f f f ff f f f f ff f f f ff f f f ff f f f f ff f f f ff f f f ff f f f f ff f f f ff f f f ff f f f f ff f f f ff f f f = h h 2 4 + 4h + h
=
@
5 2+h
+
3
@
8 @ 10 + 3
f f f f f f ff f f f ff f f f f ff f f f ff f f f ff f f f f ff f f f ff f f f ff f f f f ff f f f ff f f f ff f f f f ff f f f ff f f f ff f f f f ff f f f ff f f f ff f f f f ff f f f ff f f f ff f f f f ff f f f ff f f f h
8 + 8h + 2h
=
2
2
@ 10 @ 5h + 3 @ @ 2 + 3 f f f f f f ff f f f ff f f f f ff f f f ff f f f ff f f f f ff f f f ff f f f ff f f f f ff f f f ff f f f ff f f f f ff f f f ff f f f ff f f f f ff f f f ff f f f ff f f f f ff f f f ff f f
h
2
=
+ @ hf 1 +f 3ff 2f f f ff f ff f f f ff f fhf ff ff f f f ff f1f ff ff f = 3 + 2h h
A function can also be defined differently for different sets of values of x.
` a `a
Example : Find f @ 3 andf 4 where the function f(x) is defined as follows : x + 1 for for x <0 f x = f or x ≥ 0 1 @ 2x
`a V
Solution : Since @ 3<0, Since 4 ≥ 0,
`a `a
using f x using f x
= x +
1
= 1 @ 2x
` a `a `a
we ge get f @ 3 we get f 4
=@3 +
=1@2
4
1 =@2 = 1 @8 =@7
GRAPHS OF FUNCTIONS f is the set of all points ( x,y) such that y = f(x) The graph of f Or The graph of f is the graph of the equation y =f(x)
`a
To draw the graph of y = f x we make a table consisting two columns : one for x and one for y [or f(x)]. We take different values of x and calculate the corresponding values of y [or f(x)]. Then we plot the points on a graph paper and join them with a smooth curve to get the graph of the function .
`a
Example : Draw the the graph of f x
= x
3
Solution : We first make a table. We take different values of x and calculate the corresponding values of y [or f(x)]. Then we plot the points on a graph paper and join them with a smooth curve to get the graph x :
-2
y or f(x) :
-8
3f f f 2 27f f f f f f f @ 8 @
-1 -1
1f f f 2 1f f f @ 8 @
0 0
1f f f 2 1f f f 8
1 2
3f f f 2 27f f f f f f f 8
2 8
VERTICAL LINE TEST:
If the graph of the relation ( between x and y ) is given, we can use the vertical line test to check whether it is a graph of a function or not i.e. whether y is a function of x or not.
In the graph above, each vertical line, wherever we draw it, will intersect the graph in, at most, one point. This means for any given value of x-coordinate, we shall have maximum one y-coordinate. So y is a function of x.
In the graph above, we have at least one vertical line, which has intersected the graph in more than one point. This means for at least one value of x-coordinate, we have more than one y-coordinate. So y is not unique for some value of x. Using the definition of function, y is not a function of x.
COMBINING FUNCTIONS Sometimes we need to combine two or more functions to make a new function. The definition of these combinations by summation, subtraction, multiplication & division and composition are as follows : 1A 2A 3A 4A 5A
b b b f
c` a ` a ` a c` a ` a ` a c` a ` a ` a g` a `` aa ` a `a `a b c` a b ` ac `a f + g x
= f
x
+
g x
f @ g x
= f
x
@
g x
fg x
= f
f f f f f f x g
=
x Ag x
f f xf f fff f f f f f f f f where g x g x
≠0
Comp Compos osiition tion of two func functi tion onss f x and g x is deno denote ted d as f N g and and is defi define ned d by :
f N g x
=
f g x
b c` a
This This is also also cal called functi function on of a functi function on A The domai main of thi this new fun functio tion f N g x is the the set of all x , such tha that g x fal falls in the domai domain n of the firs firstt functi function on f A
Example :
`a b c` a
Giv en f x
= 2 x +
f N g x
and
1, g x
c` a ` a ` a ` c` a ` a ` c` a ` a ` a ` g` a c` a b ` ac ` c` a b ` ac `
f + g x
= f
f @ g x
=
fg x
= f
f f f f f f x g
=
x
2 x
g x
+
+
1
x Ag x
=
=
f g x
g N f x
=g
2 x
+
a` a
1
x
+
4
=
2 x
+
1 A x
+
4
g 2 x
+
=
x
f @ g x ,
x
+
+
4
= 3 x +
= x
@
3
+
4
= 2 x
a ` a a` a =
1
2 x
=
+
2 x
4
+
1
+
x
+
4
1 = 2 x
+
+
1 x
+
9
4 = 2 x
+
x f + fff3ffff f Example : If f x = x @ 4, an a nd g x = and h x x @ 2
`a
f f f f f f x g
fg x ,
5
a a` a ` a ` a
@
= f
f x
4, find f + g x ,
A
2 xf +f 1ffff ff ff x + 4
f N g x
b c` a b c` a b c` a f g` a
= x +
g N f x
Solution :
b b b f b b
`a b c` a
4
=
2 x 2 + 8 x + x
+
4 = 2 x 2 + 9 x
5
` a b c` a
`a
2
+
=
`a ` a
fg x , find find h 5 , h a + k A
Solution : + x f fff3ffff f = x + 2 x @ 2 x @ 2 x + 2 x + 3 = x 2 + 5 x + 6
` a b c` a ` a ` a b c ` a` a `a ` a` a ` a `a `a b c` a b ` ac f g f g b c b c` a h x
=
fg x
= f
x g x
=
=
h 5
2
=5 +
h a + k
=
x 2 @ 4
B
` a` a
5.5 + 6 = 56 a + k
2
Example : If f x
+
5 a + k
= x
2
@
+
2
6 = a 2 + 5ak + k
a nd g x 4, an
=
2
=
f g x
+
5a + 5k + 6
+ x f fff3ffffffind f N g 5 x @ 2
Solution :
f N g x
+ x f fff3fffff x @ 2
B
b ` `
c` a a a c A
2
+ x f + + fff3fffff= x f fff3fffff@ 4 = xff fff3fffffff f @4 = f 2 x @ 2 x @ 2 x @ 2 2
b
2 x + 6 x + 9 @ 4 x 2 @ 4 x + 4 + + @f @ x 2f x x x 6 9 3 22 7ff f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f+ f ff ff ff ff f ff ff f f ff ff f ff f @ = 2 4= = 2 2 x @ 4 x + 4 x @ 4 x + 4 x @ 4 x + 4
f N g 5
=
@
2
@ 3.5 + 22.5 75 + 110 7ff f f ff ff f ff ff f f ff f f f ff ff ff ff f f ff f f7f ff f= f f@ f f ff ff ff ff ff f ff ff ff ff f@ f ff f ff ff f28f f f f f f f =
5
2
@
4.5 + 4
25 @ 20 + 4
9
+
4
INVERSE FUNCTIONS
`a
`a b c` a b ` ac f g f g ` a b c` a b ` ac ` a ` a
If we take the functions f x f N g x
=
f g x
g N f x
=g
= f
f x
=
=
2 x @ 1 and and the the function ction g x
+ + x f 1 1 f f fff f f f f f f f f f x f fff f f f f f f f f f f f@ =2 1 = x 2 2
g 2 x @ 1
=
+
1
+ x f 1 f f f f f f f f f f fff f 2
=
@
1 = x
@ 2 xf 1ffff + fff ffff fff1ffff f2 xf f f f f f f = = x 2 2
In such a case, f and g are inverses of each other.
b c` a
Two Tw o fun functio tions f andg are inver veres of each oth other if f N g x
b c` a
of g an a nd if if g N f x
=
@
g
@
1
x
= f
x
A
in the the doma domaiin
x for for each x in the the domai main of f A
The The inve nverse of f is wri written tten as f
`a `a
for each ach x = x for
1
`a
@
x A In In the the above ove case we canwri anwrite f
1
`a `a x
=g
x and also also
INCREASING, DECREASING & CONSTANT FUNCTIONS 1. If for all x in an interval, the value of f(x) increases with increase in x, i.e. the graph moves upward from left to right, then the function f is called increasing function in that particular interval. If a function is increasing throughout its domain, it is referred to as an increasing function. 2. If for all x in an interval, the value of f(x) decreases with increase in x, i.e. the graph moves downward from left to right, then the function f is called decreasing function in that particular interval. If a function is decreasing throughout its domain, it is referred to as a decreasing function. 3. If for all x in an interval, the value of f(x) remains same, i.e. the graph becomes a horizontal line in that interval, then the function f is called constant function in that particular interval. If a function is constant throughout its domain, it is referred to as a constant function.
In the graph below : in the interva ntervall @ 1 , x1 i A e A @ 1 < x < x1 , the functio function n is an incr increa easi sing ng functio function, n,
` a ` a ` a
in the the inter terval val x1 ,x 2 i A e A x1 < x < x 2 , the functio function n is a decr decrea easi sing ng functio function, n, in the the inter terval val x 2 , 1 , i A e A x 2 < x < 1 , the functi function on is an incr increas easin ing g functio function nA
EVEN & ODD FUNCTIONS
` a `a ` a `a
If f @ x
= f
x for for all all x in the the doma domaiin of the func functi tion on f,
even function function A the functi function on is cal called an even
If f @ x
= @ f
x for for all all x in the the doma domaiin of the the func functi tion on f,
function A the functi function on is cal called an odd an odd function
Example : Determine whether the following functions are even or odd : a f x = 3 x 2 + 2 , b g x = x 3 @ x , c h x = x 2 + 5 x + 6
`a`a
`a `a
`a `a
Solution : 2
`a` a ` a `a ` a` a ` a ` a `a `a ` a` a ` a ` a `a a f b g
x
=3
x
=
@
@
Sinc Since, e, g
c h
x
@
=
x
@
x
@
3
x
x
x
@ @
=@g
@
@
2 = 3 x 2 + 2 A
+
2
+
and h
5
= @ x + x = @
x
@
x 3 @ x
= f
x , f is an eve even n func functi tion on A
A
x , g is an odd odd func function tion A
x
@
x
@
3
` a `a b c
Since Since,, f
+
6 = x 2 @ 5 x
≠@h
+
6A
Sinc Sincee h
` a `a x
@
≠h
x
x , the the func functi tion on f is nei neither ther eve even n nor nor odd odd A
_____________________________________________________________________
