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A function @ is defined on positive inte...

A function @ is defined on positive integers as `"@"(x) = (x + 1) (x - 1), and "@"(a) = 3`, then a =

A

`-2`

B

`-1`

C

`0`

D

`2`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem step by step, we start with the function definition and the given condition. ### Step 1: Understand the function definition The function "@" is defined as: \[ @(x) = (x + 1)(x - 1) \] This can also be expressed using the difference of squares: \[ @(x) = x^2 - 1 \] ### Step 2: Set up the equation based on the given condition We are given that: \[ @(a) = 3 \] Substituting the function definition into this equation gives: \[ a^2 - 1 = 3 \] ### Step 3: Solve for \(a^2\) To isolate \(a^2\), we add 1 to both sides of the equation: \[ a^2 - 1 + 1 = 3 + 1 \] This simplifies to: \[ a^2 = 4 \] ### Step 4: Take the square root of both sides Taking the square root of both sides gives us: \[ a = \pm 2 \] This means \(a\) can be either 2 or -2. ### Step 5: Determine the valid solution Since the function is defined on positive integers, we only consider the positive solution: \[ a = 2 \] ### Conclusion Thus, the value of \(a\) is: \[ \boxed{2} \] ---
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