Let 𝑓⁡(𝑥)= -3⁢𝑥2-2. Find and simplify the following:(a) 𝑓⁢(𝑥-5)=  (b) 𝑓⁡(𝑥-5)-𝑓⁡(-5)=

Sure, let's solve this step by step:

(a) 𝑓(𝑥-5)=

We substitute x-5 into the function f(x) = -3x^2 - 2:

f(x-5) = -3(x-5)^2 - 2 = -3(x^2 - 10x + 25) - 2 = -3x^2 + 30x - 75 - 2 = -3x^2 + 30x - 77

So, f(x-5) = -3x^2 + 30x - 77

(b) 𝑓(𝑥-5)-𝑓(-5)=

We already know that f(x-5) = -3x^2 + 30x - 77. Now we need to find f(-5):

f(-5) = -3(-5)^2 - 2 = -3(25) - 2 = -75 - 2 = -77

Now, subtract f(-5) from f(x-5):

f(x-5) - f(-5) = (-3x^2 + 30x - 77) - (-77) = -3x^2 + 30x - 77 + 77 = -3x^2 + 30x

So, f(x-5) - f(-5) = -3x^2 + 30x