Let 𝑓⁡(𝑥)= 2⁢𝑥2. Find and simplify the following:(a) 𝑓⁢(𝑥+3)=  (b) 𝑓⁡(𝑥+3)-𝑓⁡(3)=

Sure, let's solve this step by step:

(a) 𝑓⁡(𝑥+3)=

We substitute x+3 into the function f(x) = 2x^2:

f(x+3) = 2*(x+3)^2 = 2*(x^2 + 6x + 9) = 2x^2 + 12x + 18

So, f(x+3) = 2x^2 + 12x + 18

(b) 𝑓⁡(𝑥+3)-𝑓⁡(3)=

We already know f(x+3) from part (a), so we just need to find f(3) and subtract it:

f(3) = 2*(3)^2 = 2*9 = 18

Now, subtract f(3) from f(x+3):

f(x+3) - f(3) = (2x^2 + 12x + 18) - 18 = 2x^2 + 12x

So, f(x+3) - f(3) = 2x^2 + 12x