21. If a function $$ f $$ is defines as $$ f(x+2)=f(x)+f(2), x \longleftarrow-R $$ Se defines by $$ f(x)=2 x+3 $$, then find $$ f(0), f(1) $$,
i. $$ f(0)=0 $$ ii. $$ f(-2)=-f(2) $$

Question

21. If a function $$ f $$ is defines as $$ f(x+2)=f(x)+f(2), x \longleftarrow-R $$ Se defines by $$ f(x)=2 x+3 $$, then find $$ f(0), f(1) $$,
 i. $$ f(0)=0 $$ ii. $$ f(-2)=-f(2) $$

UpStudy AI · Accepted Answer

To solve the problem, we need to analyze the function $$ f $$ defined by the equation $$ f(x+2) = f(x) + f(2) $$ and the specific form $$ f(x) = 2x + 3 $$.

### Step 1: Calculate $$ f(0) $$

Using the given function $$ f(x) = 2x + 3 $$:

$$
f(0) = 2(0) + 3 = 3
$$

### Step 2: Calculate $$ f(1) $$

Using the same function:

$$
f(1) = 2(1) + 3 = 2 + 3 = 5
$$

### Step 3: Verify the conditions

#### i. Check if $$ f(0) = 0 $$

From our calculation, we found that $$ f(0) = 3 $$, which does not satisfy the condition $$ f(0) = 0 $$.

#### ii. Check if $$ f(-2) = -f(2) $$

First, we need to calculate $$ f(-2) $$ and $$ f(2) $$:

$$
f(-2) = 2(-2) + 3 = -4 + 3 = -1
$$

$$
f(2) = 2(2) + 3 = 4 + 3 = 7
$$

Now, we check if $$ f(-2) = -f(2) $$:

$$
-f(2) = -7
$$

Since $$ f(-2) = -1 $$ and $$ -f(2) = -7 $$, the condition $$ f(-2) = -f(2) $$ is not satisfied.

### Summary of Results

- $$ f(0) = 3 $$
- $$ f(1) = 5 $$
- Condition $$ f(0) = 0 $$ is **not satisfied**.
- Condition $$ f(-2) = -f(2) $$ is **not satisfied**.

Thus, the function $$ f(x) = 2x + 3 $$ does not meet the specified conditions.
$$ f(0) = 3 $$, $$ f(1) = 5 $$. The conditions $$ f(0) = 0 $$ and $$ f(-2) = -f(2) $$ are not satisfied.

21. If a function \( f \) is defines as \( f(x+2)=f(x)+f(2), x \longleftarrow-R \) Se defines by \( f(x)=2 x+3 \), then find \( f(0), f(1) \),\n i. \( f(0)=0 \) ii. \( f(-2)=-f(2) \)

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