If `f(x)=7x+12` and `f(g(x))=21x^(2)+40`, then which of the following is g(x) ?

Options

A. `21x^(2)+28`
B. `21x^(2)`
C. `7x^(2)+4`
D. `3x^(2)+4`

Explanation

To solve the problem, we need to find the function \( g(x) \) given that \( f(x) = 7x + 12 \) and \( f(g(x)) = 21x^2 + 40 \).

### Step-by-Step Solution:

1. **Substitute \( g(x) \) into \( f(x) \)**:
\[ f(g(x)) = 7g(x) + 12 \]
This is because \( f(x) = 7x + 12 \).

2. **Set the equation equal to the given expression**:
\[ 7g(x) + 12 = 21x^2 + 40 \]

3. **Isolate \( 7g(x) \)**:
Subtract 12 from both sides:
\[ 7g(x) = 21x^2 + 40 - 12 \]
Simplifying the right side gives:
\[ 7g(x) = 21x^2 + 28 \]

4. **Divide both sides by 7**:
\[ g(x) = \frac{21x^2 + 28}{7} \]
Simplifying this expression:
\[ g(x) = 3x^2 + 4 \]

5. **Final answer**:
Thus, the function \( g(x) \) is:
\[ g(x) = 3x^2 + 4 \]

### Conclusion:
The correct option for \( g(x) \) is \( 3x^2 + 4 \).

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