Let's say we have a function \(f: R \rightarrow R\) defined as \(f(x) = 3x - 7\). Is the function surjective (onto)?

Step 1 ：Step 1: A function is surjective (onto) if for every element in the codomain, there exists an element in the domain such that the function maps this element to the given element in the codomain. In other words, for our function \(f\), for every \(y\) in \(R\), there exists an \(x\) in \(R\) such that \(f(x) = y\).

Step 2 ：Step 2: Let's express \(y\) in terms of \(x\) using our function definition: \(f(x) = y = 3x - 7\).

Step 3 ：Step 3: Solve the above equation for \(x\): \(x = \frac{y + 7}{3}\).

Step 4 ：Step 4: The solution for \(x\) is a real number for every real \(y\), as the set of real numbers is closed under addition and division (except division by zero, which is not the case here).

Step 5 ：Step 5: Therefore, for every \(y\) in \(R\), there exists an \(x\) in \(R\) such that \(f(x) = y\). This means our function \(f\) is surjective (onto).