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

1. Break Down the Problem

We have the function $f(x) = -3x^2 - 2$. We need to solve two parts:

1. Find $f(x - 5)$.
2. Calculate $f(x - 5) - f(-5)$.

2. Relevant Concepts

The function $f(x) = -3x^2 - 2$ is a quadratic function. We will substitute $x - 5$ and $-5$ into the function to get the desired values.

3. Analysis and Detail

Part (a) $f(x - 5)$

To find $f(x - 5)$: $f(x-5) = -3(x-5)^2 - 2$ Calculating $(x-5)^2$: $(x-5)^2 = x^2 - 10x + 25$ Then substituting back: $f(x-5) = -3(x^2 - 10x + 25) - 2$ Distributing $-3$: $f(x-5) = -3x^2 + 30x - 75 - 2$ Combining like terms: $f(x-5) = -3x^2 + 30x - 77$

Part (b) $f(x - 5) - f(-5)$

First, we need to compute $f(-5)$: $f(-5) = -3(-5)^2 - 2 = -3(25) - 2 = -75 - 2 = -77$ Now, substituting both results: $f(x - 5) - f(-5) = (-3x^2 + 30x - 77) - (-77)$ This simplifies to: $f(x - 5) - f(-5) = -3x^2 + 30x - 77 + 77$ Thus: $f(x - 5) - f(-5) = -3x^2 + 30x$

4. Verify and Summarize

We derived $f(x - 5) = -3x^2 + 30x - 77$ and $f(-5) = -77$. Finally, we computed $f(x - 5) - f(-5) = -3x^2 + 30x$ successfully.

Final Answer

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