If `f={(1,1),(2,3),(3,5),(4,7)}`, is a linear function from Z to Z , then find f(x).

Explanation

To find the linear function \( f(x) \) given the set of points \( f = \{(1,1), (2,3), (3,5), (4,7)\} \), we can follow these steps:

### Step 1: Assume the form of the linear function
A linear function can be expressed in the form:
\[ f(x) = ax + b \]
where \( a \) and \( b \) are constants that we need to determine.

### Step 2: Set up equations using the given points
Using the points provided in the set, we can create equations.

1. From the point \( (1, 1) \):
\[ f(1) = a(1) + b = 1 \quad \Rightarrow \quad a + b = 1 \quad \text{(Equation 1)} \]

2. From the point \( (2, 3) \):
\[ f(2) = a(2) + b = 3 \quad \Rightarrow \quad 2a + b = 3 \quad \text{(Equation 2)} \]

### Step 3: Solve the system of equations
Now we have a system of two equations:
1. \( a + b = 1 \)
2. \( 2a + b = 3 \)

We can subtract Equation 1 from Equation 2 to eliminate \( b \):
\[ (2a + b) - (a + b) = 3 - 1 \]
This simplifies to:
\[ 2a - a = 2 \quad \Rightarrow \quad a = 2 \]

### Step 4: Substitute back to find \( b \)
Now that we have \( a \), we can substitute it back into Equation 1 to find \( b \):
\[ 2 + b = 1 \quad \Rightarrow \quad b = 1 - 2 = -1 \]

### Step 5: Write the final function
Now that we have both \( a \) and \( b \), we can write the function:
\[ f(x) = 2x - 1 \]

### Final Answer
Thus, the linear function is:
\[ f(x) = 2x - 1 \]

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