If `f:R to R` is defined as:
`f(x)= {{:(2x+1, "if",x gt 2),(x^(2)-1,"if",-2 lt x lt 2),(2x,"if",x lt -2):}`
then evaluate the following:
(i) `f(1)`
(ii) `f(5)`
(iii) `f(-3)`

Answer

To solve the problem, we need to evaluate the function \( f(x) \) at three different values: \( f(1) \), \( f(5) \), and \( f(-3) \). The function is defined piecewise as follows:

1. \( f(x) = 2x + 1 \) if \( x > 2 \)
2. \( f(x) = x^2 - 1 \) if \( -2 < x < 2 \)
3. \( f(x) = 2x \) if \( x < -2 \)

Now, let's evaluate each case step by step.

### Step 1: Evaluate \( f(1) \)

1. Identify the interval for \( x = 1 \):
- Since \( -2 < 1 < 2 \), we use the second piece of the function: \( f(x) = x^2 - 1 \).

2. Substitute \( x = 1 \) into the function:
\[ f(1) = 1^2 - 1 = 1 - 1 = 0 \]

### Step 2: Evaluate \( f(5) \)

1. Identify the interval for \( x = 5 \):
- Since \( 5 > 2 \), we use the first piece of the function: \( f(x) = 2x + 1 \).

2. Substitute \( x = 5 \) into the function:
\[ f(5) = 2(5) + 1 = 10 + 1 = 11 \]

### Step 3: Evaluate \( f(-3) \)

1. Identify the interval for \( x = -3 \):
- Since \( -3 < -2 \), we use the third piece of the function: \( f(x) = 2x \).

2. Substitute \( x = -3 \) into the function:
\[ f(-3) = 2(-3) = -6 \]

### Final Results

- \( f(1) = 0 \)
- \( f(5) = 11 \)
- \( f(-3) = -6 \)

### Summary of Answers:
- \( f(1) = 0 \)
- \( f(5) = 11 \)
- \( f(-3) = -6 \)