1. `f(x)=x(x+1)(2x+1)`. Find `f(x)-f(x-1)`
Solution:
`f(x)=x(x+1)(2x+1)`
`f(x)-f(x-1)=?`
`f(x)``=x(x+1)(2x+1)`
`=x(2x^(2)+3x+1)`
`=2x^(3)+3x^(2)+x`
`f(x-1)``=(x-1)((x-1)+1)(2(x-1)+1)`
`=(x-1)(x)(2x-1)`
`=(x-1)x(2x-1)`
`=x(2x^(2)-3x+1)`
`=2x^(3)-3x^(2)+x`
Now `f(x)-f(x-1)`
`(2x^(3)+3x^(2)+x)-(2x^(3)-3x^(2)+x)`
`=(2x^(3)+3x^(2)+x)-(2x^(3)-3x^(2)+x)`
`=6x^(2)`
2. `f(x)=x^2-x`. Find `f(x+1)-f(x)`
Solution:
`f(x)=x^2-x`
`f(x+1)-f(x)=?`
`f(x+1)=(x+1)^2-(x+1)`
`=(x+1)^2-x-1`
`=(x^2+2x+1)-x-1`
`=x^2+x`
Now `f(x+1)-f(x)`
`=(x^2+x)-(x^2-x)`
`=x^2+x-x^2+x`
`=2x`
3. `f(x)=2x-3`. Find `f(0)+f(1)+f(2)`
Solution:
`f(x)=2x-3`
`f(0)+f(1)+f(2)=?`
`f(0)=2*0-3`
`=0-3`
`=-3`
`f(1)=2*1-3`
`=2-3`
`=-1`
`f(2)=2*2-3`
`=4-3`
`=1`
Now `f(0)+f(1)+f(2)`
`=(-3)+(-1)+(1)`
`=-3-1+1`
`=-3`
This material is intended as a summary. Use your textbook for detail explanation.
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