Data Sufficiency

GMATH practice exercise (Quant Class 14)

What is the value of F(-1)-F(1) ?

(1) F(x) = x^2 , for all x
(2) F(x+1) = F(x) + 2x + 1, for all x

Answer: [spoiler]____(D)__[/spoiler]

fskilnik@GMATH wrote:GMATH practice exercise (Quant Class 14)

What is the value of F(-1)-F(1) ?

(1) F(x) = x^2 , for all x
(2) F(x+1) = F(x) + 2x + 1, for all x

$$? = F\left( { - 1} \right) - F\left( 1 \right)$$
$$\left( 1 \right)\,\,F\left( x \right) = {x^2}\,,\,\,{\rm{for}}\,\,{\rm{all}}\,\,x\,\,\,\, \Rightarrow \,\,\,\,F\left( x \right) = F\left( { - x} \right)\,,\,\,{\rm{for}}\,\,{\rm{all}}\,\,x\,\,\,\,\mathop \Rightarrow \limits^{x\, = \,1} \,\,\,\,\,F\left( 1 \right) = F\left( { - 1} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = 0$$

$$\left( 2 \right)\,\,F\left( {x + 1} \right) - F\left( x \right) = 2x + 1\,,\,{\rm{for}}\,\,{\rm{all}}\,\,x\,\,\,\left( * \right)$$
$$\left. \matrix{
{\rm{Take}}\,\,x = - 1\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,F\left( 0 \right) - F\left( { - 1} \right) = - 2 + 1\,\,\, \hfill \cr
{\rm{Take}}\,\,x = 0\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,F\left( 1 \right) - F\left( 0 \right) = 1 \hfill \cr} \right\}\,\,\,\,\,\mathop \Rightarrow \limits^{\left( + \right)} \,\,\,\,\,F\left( 1 \right) - F\left( { - 1} \right) = 1 + \left( { - 2 + 1} \right) = 0\,\,\,\,\, \Rightarrow \,\,\,\,\,? = - 0 = 0$$


The correct answer is therefore (D).


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.