Determine if the relation \(f(x) = 3x^2 - 2x + 1\) is injective (one-to-one).
Answer
Step 5: Therefore, we must have \(a = b\). This means the function is injective (one-to-one).
Step 1 ：Step 1: Assume that \(f(a) = f(b)\) for some \(a, b\) in the domain of \(f\). We have \(3a^2 - 2a + 1 = 3b^2 - 2b + 1\).
Step 2 ：Step 2: Simplify the equation to \(3a^2 - 2a = 3b^2 - 2b\), which can be rewritten as \(3(a^2 - b^2) = 2(a-b)\).
Step 3 ：Step 3: Factor both sides to get \(3(a - b)(a + b) = 2(a - b)\).
Step 4 ：Step 4: If \(a \neq b\), we can divide both sides by \(a - b\) and get \(3(a + b) = 2\), which is a contradiction because \(3(a + b)\) can never equal to 2 for any \(a, b\).
Step 5 ：Step 5: Therefore, we must have \(a = b\). This means the function is injective (one-to-one).
