2. Fie functiile f, g : R −→ R, definite prin f (x) = x − 2, g(x) = 2 − x, ∀x ∈ R(a) Sa se calculeze (f ◦ g)(1)(b) Sa se calculeze sum100∑k=1( (f ◦ g)(k) )2

1. Break Down the Problem

We need to solve two parts:

1. Calculate $(f \circ g)(1)$.
2. Calculate the sum $\sum_{k=1}^{100} (f \circ g)(k)^2$.

2. Relevant Concepts

• The composition of functions $f \circ g$ means we will apply $g$ first, and then apply $f$ to the result.
• The given functions are:
  • $f(x) = x - 2$
  • $g(x) = 2 - x$

3. Analysis and Detail

(a) Calculate $(f \circ g)(1)$:

1. Compute $g(1)$: $g(1) = 2 - 1 = 1$
2. Now compute $f(g(1)) = f(1)$: $f(1) = 1 - 2 = -1$ Thus, $(f \circ g)(1) = -1$.

(b) Calculate $\sum_{k=1}^{100} (f \circ g)(k)^2$:

1. First, find $(f \circ g)(k)$:

   • Compute $g(k)$: $g(k) = 2 - k$
   • Now compute $f(g(k)) = f(2 - k)$: $f(2 - k) = (2 - k) - 2 = -k$ Therefore, $(f \circ g)(k) = -k$.

2. Now compute the sum: $\sum_{k=1}^{100} (f \circ g)(k)^2 = \sum_{k=1}^{100} (-k)^2 = \sum_{k=1}^{100} k^2$ The formula for the sum of the squares of the first $n$ natural numbers is: $\sum_{k=1}^{n} k^2 = \frac{n(n + 1)(2n + 1)}{6}$ For $n = 100$: $\sum_{k=1}^{100} k^2 = \frac{100(100 + 1)(2 \times 100 + 1)}{6} = \frac{100 \times 101 \times 201}{6}$ Calculate this step-by-step:

   • $100 \times 101 = 10100$
   • $2 \times 100 + 1 = 201$
   • $10100 \times 201 = 2030100$
   • $2030100 \div 6 = 338350$

4. Verify and Summarize

• Part (a) results in $(f \circ g)(1) = -1$.
• Part (b) results in the sum $\sum_{k=1}^{100} (f \circ g)(k)^2 = 338350$.

Final Answer

• (a) $(f \circ g)(1) = -1$
• (b) $\sum_{k=1}^{100} (f \circ g)(k)^2 = 338350$