Paper 4: Fundamentals of Business Mathematics & Statistic

FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 3.5

2

2x ,

x 1

=

−^1 ≤ x < 3
Find f (– 1), f(0), f (1/2), f (2).
f( – 1) = 5– 2 (–1) – 1 (since – 1 lies in – 1 ≤ x < 0)
= 5^2 – 1 = 25 – 1 = 24.

Points 0, 21 lie in the second interval ; so,

f 0( )=0 20 2−− =1,

(^12)
f^147.
2 1 2 6
2

−

= =

−

Now 2 lies in the third interval f(2) =2 12.2 4 4 (^2) − =4 1 3− =.
Example22 : If f(x) = eax + b. Prove that eb f(x + y) = f (x). f(y)
eb f(x + y) = =ee.e +++++ bayaxbb)yx(ab
= eax + b. eay + b = f(x). f(y). Hence proved.
Example23 : If f(x) = x – a, q(x) = x + a then show that
{f(x)}^2 – {q(x)}^2 = – 2a {f(x) + q(x)}
L.H.S. = (x – a)^2 – (x + a)^2 = x^2 + a^2 – 2ax – (x^2 + a^2 + 2ax) = – 4ax
R.H.S. = – 2a {x – a + x + a} = – 2a. 2x = – 4ax.
SELF EXAMINATION

1. If f(x) = (x – 1) (x – 2 (x – 3) find the values of
   (i) f(1) (ii) f(2) (iii) f(3) (vi)f(0). [Ans. 0, 0, 0, – 6]


2. If f(x) = | x | + x, find whether f (3) and f(– 3) are equal? [Ans. No]


3. Given f(x) = x, and F(x)
   x^2
   = x is F(x) = f(x) always? [Ans. equal, except x = 0]


4. (i) If f(x) = x^2 + 2x^4 verify f(x) = f (– x).
   (ii) If f(x) = x + 2x^3 verify f(x) = – f (– x)


5. If y = 5 for every value of x, can y be regarded as a function of x? [Ans. Yes]


6. f(x) = x + | x |, are f(4) and f(– 4) equal? [Ans. No]


7. If f x b( )= b a a bx a x b−− +a ,−− then f(a) + f(b) = f(a + b).


8. If f(x) = x^2 – x, then prove that f(h+1) = f(– h)


9. If φ(x) a= a b b ax b x a−− +b , a b−− ( ≠ ) obtain φ (a), φ (b) and φ (a + b) and then verify

φ (a) + φ (b) = φ (a + b).

10. If y f(x)= =cx aax b+− prove that φ(y) = x.