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Let ๐‘“โก(๐‘ฅ)= 2โข๐‘ฅ2. Find and simplify the following:(a) ๐‘“โข(๐‘ฅ+3)= ย (b) ๐‘“โก(๐‘ฅ+3)-๐‘“โก(3)=

Question

Let f(x)=2x2 f(x) = 2x^2 . Find and simplify the following:

(a) f(x+3)= f(x+3) =
(b) f(x+3)โˆ’f(3)= f(x+3) - f(3) =

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Solution

Sure, let's solve this step by step:

(a) ๐‘“โก(๐‘ฅ+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) ๐‘“โก(๐‘ฅ+3)-๐‘“โก(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

This problem has been solved

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