Thread: function is defined by folliowing rule question.

1. function is defined by folliowing rule question.

a function is defiend by the following rule:

f (x) is
0 if x lesser than/equal to -3
-1 if -3<x<0
x if x is greater then or equal to 0

find i) F (-3) + f (-2) + f (2)
ii) f (a ^2)

another question:
4 ^ (x+1)/ ( 4 ^(x + 2) – 4 ^ (x) )

and another.

x-1 is a factor of x ^ 3 - (2 X ^ 2) + px + 6
find the value of P

and another
fully factorise the polynomial P(X) = X ^4 - 7x^3 + 12x^2 + 4x -16
given that x = 2 is a double root of P(x)

2. Originally Posted by thcbender
a function is defiend by the following rule:

f (x) is
0 if x lesser than/equal to -3
-1 if -3<x<0
x if x is greater then or equal to 0

find i) F (-3) + f (-2) + f (2)
ii) f (a ^2)
(i) $\displaystyle f(-3)+f(-2)+f(2)=0+(-1)+2=1$
(ii) Since $\displaystyle a^{2}\geq0$, so $\displaystyle f(a^{2})=a^{2}$

Originally Posted by thcbender
another question:
4 ^ (x+1)/ ( 4 ^(x + 2) – 4 ^ (x) )
$\displaystyle \frac{4^{x+1}}{4^{x+2}-4^{x}}=\frac{4(4^{x})}{4^{2}4^{x}-4^{x}}$
$\displaystyle =\frac{4(4^{x})}{4^{x}(4^{2}-1)}=\frac{4}{4^{2}-1}=...$

Originally Posted by thcbender

x-1 is a factor of x ^ 3 - (2 X ^ 2) + px + 6
find the value of P
Since $\displaystyle (x-1)$ is a factor, by factor theorem, $\displaystyle f(1)=0$, where $\displaystyle f(x)=x^{3}-2x^{2}+px+6$
So $\displaystyle 1^{3}-2(1)^{2}+p(1)+6=0$...solve it you will get your p

Originally Posted by thcbender
fully factorise the polynomial P(X) = X ^4 - 7x^3 + 12x^2 + 4x -16
given that x = 2 is a double root of P(x)
Since $\displaystyle x=2$ is a double root, so $\displaystyle (x-2)^{2}$ is a factor, get the other quadratic factor by either long division, comparison or synthetic division. You will get $\displaystyle f(x)=(x-2)^{2}(x^{2}-3x-4)=(x-2)^{2}(x-4)(x+1)$