What value must be excluded from the domain of `((f)/(g))(x)` if `f(x)=3x^(2)-4x+1` and `g(x)=3x^(2)-3`?

Options

A. 0
B. 1
C. 3
D. both `+-1`

Explanation

To determine what value must be excluded from the domain of the function \(\frac{f}{g}(x)\) where \(f(x) = 3x^2 - 4x + 1\) and \(g(x) = 3x^2 - 3\), we need to follow these steps:

### Step 1: Identify the functions
We have:
- \(f(x) = 3x^2 - 4x + 1\)
- \(g(x) = 3x^2 - 3\)

### Step 2: Write the composite function
The composite function can be expressed as:
\[ \frac{f}{g}(x) = \frac{f(x)}{g(x)} = \frac{3x^2 - 4x + 1}{3x^2 - 3} \]

### Step 3: Determine when the denominator is zero
To find the values that must be excluded from the domain, we need to find when the denominator \(g(x)\) is equal to zero:
\[ g(x) = 3x^2 - 3 = 0 \]

### Step 4: Solve for \(x\)
Setting the denominator equal to zero:
\[ 3x^2 - 3 = 0 \]
Dividing both sides by 3:
\[ x^2 - 1 = 0 \]
Factoring the equation:
\[ (x - 1)(x + 1) = 0 \]
Setting each factor to zero gives us:
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]
\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]

### Step 5: Conclusion
The values \(x = 1\) and \(x = -1\) make the denominator zero, and thus must be excluded from the domain of \(\frac{f}{g}(x)\).

### Final Answer
The values that must be excluded from the domain of \(\frac{f}{g}(x)\) are \(x = 1\) and \(x = -1\).

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