Question

If F(x)=√3x+2 find (a) f(a+h) -f(a) over h
(b) if F:x mapping x^2 over 1+x^2 find f(x2)- f(x1)

Answer 1

(a)

$${f(a+h)-f(a) \over n}={\sqrt{3(a+h)+2}-\sqrt{3a+2} \over h}=$$

$$={3a+3h+2-3a-2\over \ h(\sqrt{3(a+h)+2}+\sqrt{3a+2})}={3\over \ \sqrt{3(a+h)+2}+\sqrt{3a+2}}$$

(b)

$$f(x_2)-f(x_1)={x_2^2 \over x_2^2+1}-{x_1^2 \over x_1^2+1}=$$

$$={x_2^2x_1^2+x_2^2-x_2^2x_1^2-x_1^2 \over (x_1^2+1)(x_2^2+1)}={x_2^2-x_1^2 \over (x_1^2+1)(x_2^2+1)}=$$

$$={(x_2-x_1)(x_2+x_1) \over (x_1^2+1)(x_2^2+1)}$$