The domain for which the functions defined by `f(x)=6x^(2)+1 and g(x)=11-7x` are equal is

Options

A. `{-1,(2)/(3)}`
B. `{3,(5)/(6)}`
C. `{-2,(5)/(6)}`
D. `{2,(2)/(3)}`

Answer

To find the domain for which the functions defined by \( f(x) = 6x^2 + 1 \) and \( g(x) = 11 - 7x \) are equal, we need to set the two functions equal to each other and solve for \( x \).

### Step-by-Step Solution:

1. **Set the functions equal to each other:**
\[ f(x) = g(x) \]
This gives us:
\[ 6x^2 + 1 = 11 - 7x \]

2. **Rearrange the equation:**
Move all terms to one side of the equation:
\[ 6x^2 + 7x + 1 - 11 = 0 \]
Simplifying this, we get:
\[ 6x^2 + 7x - 10 = 0 \]

3. **Factor the quadratic equation:**
We need to factor \( 6x^2 + 7x - 10 \). To do this, we look for two numbers that multiply to \( 6 \times (-10) = -60 \) and add to \( 7 \). The numbers \( 12 \) and \( -5 \) work:
\[ 6x^2 + 12x - 5x - 10 = 0 \]
Grouping the terms:
\[ (6x^2 + 12x) + (-5x - 10) = 0 \]
Factoring by grouping:
\[ 6x(x + 2) - 5(x + 2) = 0 \]
This can be factored as:
\[ (x + 2)(6x - 5) = 0 \]

4. **Solve for \( x \):**
Set each factor to zero:
\[ x + 2 = 0 \quad \text{or} \quad 6x - 5 = 0 \]
Solving these gives:
\[ x = -2 \quad \text{or} \quad x = \frac{5}{6} \]

5. **Conclusion:**
The values of \( x \) for which the functions \( f(x) \) and \( g(x) \) are equal are \( x = -2 \) and \( x = \frac{5}{6} \). Therefore, the domain for which the functions are equal is:
\[ \{ -2, \frac{5}{6} \} \]