Let f = {(1, 1),(2,3),(0,-1), (-1,-3)} be a linear function from Z into Z, then f(x) =

Options

A. 2x - 1
B. 2x
C. 2x + 1
D. `-2x` + 1

Answer

To find the linear function \( f(x) \) given the set of ordered pairs \( f = \{(1, 1), (2, 3), (0, -1), (-1, -3)\} \), we will follow these steps:

### Step 1: Identify the ordered pairs
The ordered pairs given are:
- \( (1, 1) \)
- \( (2, 3) \)
- \( (0, -1) \)
- \( (-1, -3) \)

We can denote these pairs as \( (x, y) \), where \( x \) is the input and \( y \) is the output.

### Step 2: Assume a linear function
A linear function can be expressed in the form:
\[ f(x) = mx + b \]
where \( m \) is the slope and \( b \) is the y-intercept.

### Step 3: Calculate the slope \( m \)
To find the slope \( m \), we can use any two points. Let's use the points \( (1, 1) \) and \( (2, 3) \):
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 1}{2 - 1} = \frac{2}{1} = 2 \]

### Step 4: Find the y-intercept \( b \)
Now that we have \( m = 2 \), we can use one of the points to find \( b \). Let's use the point \( (1, 1) \):
\[ 1 = 2(1) + b \implies 1 = 2 + b \implies b = 1 - 2 = -1 \]

### Step 5: Write the function
Now we can write the linear function:
\[ f(x) = 2x - 1 \]

### Step 6: Verify with all points
We can verify this function with all the given points:
- For \( x = 1 \): \( f(1) = 2(1) - 1 = 1 \) (matches)
- For \( x = 2 \): \( f(2) = 2(2) - 1 = 3 \) (matches)
- For \( x = 0 \): \( f(0) = 2(0) - 1 = -1 \) (matches)
- For \( x = -1 \): \( f(-1) = 2(-1) - 1 = -3 \) (matches)

All points match, confirming our function.

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