Is `g={(1,1),(2,3),(3,5,),(4,7)}` a function? If this is described by the formula, `g(x)=alphax+beta,` then what values should be assigned to `alphaa n dbeta?`

Options

A. `alpha = 1, beta = 1`
B. `alpha = 2, beta = - 1`
C. `alpha = 1, beta = - 2`
D. `alpha = - 2, beta = - 1`

Answer

To determine if \( g = \{(1,1), (2,3), (3,5), (4,7)\} \) is a function and to find the values of \( \alpha \) and \( \beta \) in the equation \( g(x) = \alpha x + \beta \), we can follow these steps:

### Step 1: Check if \( g \) is a function
A relation is a function if every input (or x-value) corresponds to exactly one output (or y-value).

In the given set:
- For \( x = 1 \), \( g(1) = 1 \)
- For \( x = 2 \), \( g(2) = 3 \)
- For \( x = 3 \), \( g(3) = 5 \)
- For \( x = 4 \), \( g(4) = 7 \)

Since every x-value has a unique y-value, \( g \) is indeed a function.

### Step 2: Set up the equations
We can express the function in the form \( g(x) = \alpha x + \beta \). We will use two points from the function to create equations. Let's use the points \( (1, 1) \) and \( (2, 3) \).

1. From the point \( (1, 1) \):
\[ 1 = \alpha(1) + \beta \quad \text{(Equation 1)} \]
This simplifies to:
\[ \alpha + \beta = 1 \]

2. From the point \( (2, 3) \):
\[ 3 = \alpha(2) + \beta \quad \text{(Equation 2)} \]
This simplifies to:
\[ 2\alpha + \beta = 3 \]

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

We can solve these equations by elimination.

Subtract Equation 1 from Equation 2:
\[ (2\alpha + \beta) - (\alpha + \beta) = 3 - 1 \]
This simplifies to:
\[ 2\alpha - \alpha = 2 \implies \alpha = 2 \]

### Step 4: Substitute \( \alpha \) back to find \( \beta \)
Now that we have \( \alpha = 2 \), we can substitute this value back into Equation 1:
\[ 2 + \beta = 1 \]
Solving for \( \beta \):
\[ \beta = 1 - 2 = -1 \]

### Final Answer
Thus, the values are:
\[ \alpha = 2, \quad \beta = -1 \]