I need help for these 2 math questions

Question

23. If f(x)  = x^2  +3 find (f(x +h)  -f(x))/h

 

22. Let f be f(x)  =  (3x^2  +8x +8)/(x^2  -4) find the vertical and horizontal asymptotes

Answer 1

(22).

The function is f(x) = (3x² + 8x + 8) / (x² - 4).

In the above function, numerator function [N(x) = 3x² + 8x + 8] and denominator function [D(x) = x² - 4] have no common factors.

The graph of f(x) has vertical asymptotes at the zeros of denominator function.

D(x) = x² - 4 = 0 ⇒ x = ± 2.

The vertical asymptotes x = ± 2.

 

The graph of f(x) function has one or no horizontal asymptote determined by comparing the degrees of numerator function and denominator function.

Degree of numerator function = 1 = Degree of denominator function.

Therefore, horizontal asymptote is y = a_n/b_n = leading coefficient of numerator function / leading coefficient of denominator function = 3/1 = 3.

The horizontal asymptote y = 3.

 

Answer 2

f(x)  = x^2  +3

f(x+h) = (x+h)^2 + 3

          = x^2 + 2xh +h^2 +3        ((a+b)^2  =a^2 +2ab +b^2)

          = x^2 + 3 +2xh +h^2

          = f(x) + 2xh +h^2            (Since x^2  +3 = f(x))

(f(x +h)  -f(x)) = f(x) + 2xh +h^2  -f(x)

                      = 2xh +h^2

                      = h (2x + h)

(f(x +h)  -f(x))/h = h (2x + h)/h

                         = 2x +h