Question

Find and simplify the difference quotient of the form
`\fra(f(x)-f(a))/(x-a) , x\\eq a`


`f(x)=3x^2+x ,a =4`

What is the difference quotient?

Answer 1

If
`f(x)=3x^2+x`, then

`f(4) = 3\cdot4^2+4 = 52`

so that the difference quotient is

`(f(x)-f(a))/(x-a) = (3x^2+x-52)/(x-4)`

Factorize the numerator:

`3x^2+x-52 = (3x+13)(x-4)`

Then for x ≠ 4, we have

`(f(x)-f(a))/(x-a) = ((3x+13)(x-4))/(x-4) = 3x+13`

Answer 2

Answer: 3x + 13

Explanation:

`f(x)=3x^2+x\\f(a)=3a^2+a\\f(4)=3*4^2+4=52\\\\f(x)-f(4)=3x^2+x-52=(x-4)(3x+13)\\\\(f(x)-f(4))/(x-4) =(3x^2+x-52)/(x-4) \\\\if\ x \\eq 4\ then \\\\(f(x)-f(4))/(x-4) =((x-4)(3x+13))/(x-4) \\\\=3x+13\\`