Let f={(1,1),(2,3),(0,−1),(−1,−3)} be a function from Z to Z defined by f(x)=ax+b for some integers a,b. Determine a,b.

Answer

To determine the values of \( a \) and \( b \) in the function \( f(x) = ax + b \) given the pairs \( f = \{(1,1), (2,3), (0,-1), (-1,-3)\} \), we will follow these steps:

### Step 1: Set up equations using the function definition
From the pairs provided, we can create equations based on the function \( f(x) = ax + b \).

1. For the pair \( (1, 1) \):
\[ f(1) = a(1) + b = 1 \implies a + b = 1 \quad \text{(Equation 1)} \]

2. For the pair \( (0, -1) \):
\[ f(0) = a(0) + b = -1 \implies b = -1 \quad \text{(Equation 2)} \]

### Step 2: Substitute \( b \) into Equation 1
Now that we have \( b = -1 \) from Equation 2, we can substitute this value into Equation 1.

\[ a + (-1) = 1 \implies a - 1 = 1 \]

### Step 3: Solve for \( a \)
Now we solve for \( a \):

\[ a - 1 = 1 \implies a = 1 + 1 = 2 \]

### Step 4: Write down the values of \( a \) and \( b \)
Now we have determined the values:
\[ a = 2, \quad b = -1 \]

### Final Result
Thus, the values of \( a \) and \( b \) are:
\[ \boxed{(2, -1)} \]