Let's consider a function \(f: A \rightarrow B\) where \(A = \{1, 2, 3, 4\}\) and \(B = \{5, 6, 7, 8\}\). The function is defined as \(f(x) = x + 4\). Is this function surjective (onto)?
Answer
Step 4: Since every element in \(B\) is the image of at least one element in \(A\) under the function \(f\), the function \(f\) is surjective (onto).
Step 1 ：Step 1: Define the function \(f: A \rightarrow B\) where \(A = \{1, 2, 3, 4\}\) and \(B = \{5, 6, 7, 8\}\). The function is defined as \(f(x) = x + 4\).
Step 2 ：Step 2: Calculate the image of each element in the domain \(A\) under the function \(f\). For \(x = 1, 2, 3, 4\), the images are \(f(1) = 5, f(2) = 6, f(3) = 7, f(4) = 8\).
Step 3 ：Step 3: Compare the set of images with the codomain \(B\). The set of all images is \(\{5, 6, 7, 8\}\), which is exactly the set \(B\).
Step 4 ：Step 4: Since every element in \(B\) is the image of at least one element in \(A\) under the function \(f\), the function \(f\) is surjective (onto).
