Tim T. answered • 04/21/19

Math: K-12th grade to Advanced Calc, Ring Theory, Cryptography

Greetings! Lets solve this shall we ?

We yield **the domain of the original function** f(x) = 3x^{2}-4x+5 **is** **all real numbers** or **(-∞, ∞)** because it is **parabolic** in nature. Now lets find the derivative using the definition of a derivative such that,

lim f(x+h) - f(x)

h-> 0 -------------------

h

f(x+h) = 5 - 4(x+h) + 3(x+h)^{2} = 5 - 4x - 4h + 3x^{2} + 6xh + 3h^{2}

Then,

lim [5 - 4x - 4h + 3x^{2} + 6xh + 3h^{2}] - [5-4x+3x^{2}]

h->0 ----------------------------------------------------------- ; Distribute the negative into the function then simplify

h

lim -4h+6xh+3h^{2 } lim h( -4 + 6x + 3h) lim

h->0 --------------------- = h->0 ------------------------ = h->0 (-4+6x+3h) = **6x-4**

h h

Since (6x-4) is the derivative of the function, the domain of it is also **all real numbers **or **(-∞, ∞) **because the derivative is **linear**.

I hope this helped!