For which Domain, the functions `f(x) = 2x^2-1` and `g(x)=1-3x` are equal to

Options

A. `[2,-1//2]`
B. `[-2,1//2]`
C. `[1,2]`
D. `[-2,-1//2]`

Explanation

To determine the domain for which the functions \( f(x) = 2x^2 - 1 \) and \( g(x) = 1 - 3x \) are equal, we will follow these steps:

### Step 1: Set the functions equal to each other
We start by equating the two functions:
\[ f(x) = g(x) \]
This gives us:
\[ 2x^2 - 1 = 1 - 3x \]

### Step 2: Rearrange the equation
Next, we rearrange the equation to bring all terms to one side:
\[ 2x^2 + 3x - 1 - 1 = 0 \]
This simplifies to:
\[ 2x^2 + 3x - 2 = 0 \]

### Step 3: Factor the quadratic equation
Now, we will factor the quadratic equation \( 2x^2 + 3x - 2 = 0 \). We look for two numbers that multiply to \( 2 \times -2 = -4 \) and add up to \( 3 \). The numbers \( 4 \) and \( -1 \) work:
\[ 2x^2 + 4x - x - 2 = 0 \]
Grouping the terms:
\[ (2x^2 + 4x) + (-x - 2) = 0 \]
Factoring by grouping:
\[ 2x(x + 2) - 1(x + 2) = 0 \]
This gives us:
\[ (2x - 1)(x + 2) = 0 \]

### Step 4: Solve for \( x \)
Now, we set each factor equal to zero:
1. \( 2x - 1 = 0 \) leads to:
\[ 2x = 1 \implies x = \frac{1}{2} \]
2. \( x + 2 = 0 \) leads to:
\[ x = -2 \]

### Step 5: Identify the domain
The solutions \( x = \frac{1}{2} \) and \( x = -2 \) are the points where the two functions are equal. Since both functions are polynomial functions, their domain is all real numbers. Thus, the domain where the functions are equal is:
\[ x = -2 \quad \text{and} \quad x = \frac{1}{2} \]

### Final Answer
The functions \( f(x) \) and \( g(x) \) are equal at the points \( x = -2 \) and \( x = \frac{1}{2} \).

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