Which of the following could be g(x) if `f(x)=5x^(2)+4` and `f(g(3))=84` ?

Options

A. `3x-10`
B. `4x-7`
C. `6x-17`
D. `x^(2)-5`

Explanation

To solve the problem, we need to find a function \( g(x) \) such that \( f(g(3)) = 84 \), where \( f(x) = 5x^2 + 4 \).

### Step-by-Step Solution:

1. **Set Up the Equation**: We know that \( f(g(3)) = 84 \). Therefore, we can set up the equation:
\[ f(g(3)) = 84 \]

2. **Substitute the Function**: Substitute the expression for \( f(x) \) into the equation:
\[ 5(g(3))^2 + 4 = 84 \]

3. **Isolate the Quadratic Term**: Subtract 4 from both sides:
\[ 5(g(3))^2 = 80 \]

4. **Divide by 5**: Divide both sides by 5 to simplify:
\[ (g(3))^2 = 16 \]

5. **Take the Square Root**: Take the square root of both sides to find \( g(3) \):
\[ g(3) = 4 \quad \text{or} \quad g(3) = -4 \]

6. **Identify Possible Functions**: Now we need to find a function \( g(x) \) such that when \( x = 3 \), \( g(3) = 4 \) or \( g(3) = -4 \).

### Checking the Options:

- **Option A**: \( g(x) = 3x - 10 \)
- \( g(3) = 3(3) - 10 = 9 - 10 = -1 \) (not correct)

- **Option B**: \( g(x) = 4x - 7 \)
- \( g(3) = 4(3) - 7 = 12 - 7 = 5 \) (not correct)

- **Option C**: \( g(x) = 6x - 17 \)
- \( g(3) = 6(3) - 17 = 18 - 17 = 1 \) (not correct)

- **Option D**: \( g(x) = x^2 - 5 \)
- \( g(3) = 3^2 - 5 = 9 - 5 = 4 \) (correct)

Thus, the function \( g(x) = x^2 - 5 \) satisfies the condition \( g(3) = 4 \).

### Conclusion:
The correct answer is:
\[ g(x) = x^2 - 5 \]