Question

Let f : R → : f(x) = x2

and g : R → : g(x) = 2x+1. Find

i) (f+g)(x) ii) (f-g)(x).​

Answer 1

EXPLANATION.

→ f : R → : f (x) = x² and g : R → : g (x) = 2x + 1

To find (1) = ( f + g) (x) (2) = (f - g) (x).

→ (1) = ( f + g) (x)

→ f(x) + g(x)

→ x² + ( 2x + 1 )

→ x² + 2x + 1

→ (2) = ( f - g) (x)

→ f(x) - g(x)

→ x² - ( 2x + 1 )

→ x² - 2x - 1

More information.

property of greatest integer function.

→ (1) = [x] ≤ x ≤ [x] + 1

→ (2) = x - 1 < [x] ≤ x

→ (3) = 0 ≤ x - [x] < 1

→ (4) = [ x + m ] = [ x] + m → if m is an integer.

→ (5) = [ x ] + [ y ] ≤ [ x + y ] ≤ [ x ] + [ y ] + 1

[ -x] = - [ x] → if x € I

[ -x ] = - [ x ] - 1 if x ¢ I

→ (6) = [ x ] + [ -x ] = 0 if x is an integer.

[ x ] + [ - x ] = -1 in other conditions.

→ (7) = if [ x ] > n → x ≥ n + 1 → n € Integer.

→ (8) = if [ x ] < n → x < n → n € integer.

→ (9) = [ x + y ] = [ x ] + [ y + x ( -x) ] for all x

y € R

→ (10) = if [ ø (x) ] ≥ I then ø (x) ≥ I

→ (11) = if [ ø (x) ] ≤ I then ø ( x) ≤ I + 1

→ (12) = x - [ x ] is the fractional part of x

→ (13) = - [ -x] is the least integer ≥ x