IF `f(x) = sqrt(x)` and the composite function f(g(x))=`sqrt(4x^2-5)`, which of the following could be g(x) ?

Options

A. `sqrt(4x^4-5)`
B. `sqrt(16x^4-25)`
C. `2x^2-25`
D. `4x^2-5`

Answer

To solve the problem, we need to find the function \( g(x) \) such that the composite function \( f(g(x)) = \sqrt{4x^2 - 5} \) holds true, given that \( f(x) = \sqrt{x} \).

### Step-by-Step Solution:

1. **Understand the Functions**:
We know that \( f(x) = \sqrt{x} \). For the composite function \( f(g(x)) \), we substitute \( g(x) \) into \( f(x) \).

\[ f(g(x)) = \sqrt{g(x)} \]

2. **Set Up the Equation**:
According to the problem, we have:

\[ f(g(x)) = \sqrt{4x^2 - 5} \]
Therefore, we can equate the two expressions:

\[ \sqrt{g(x)} = \sqrt{4x^2 - 5} \]

3. **Square Both Sides**:
To eliminate the square roots, we square both sides of the equation:

\[ g(x) = 4x^2 - 5 \]

4. **Identify the Correct Option**:
Now we need to check which of the provided options matches \( g(x) = 4x^2 - 5 \). The options given are:
- Option 1: \( \sqrt{4x^4 - 5} \)
- Option 2: \( \sqrt{16x^4 - 25} \)
- Option 3: \( 2x^2 - 25 \)
- Option 4: \( 4x^2 - 5 \)

The correct option is clearly:

\[ \text{Option 4: } 4x^2 - 5 \]

### Final Answer:
Thus, the function \( g(x) \) that satisfies the condition is:
\[ g(x) = 4x^2 - 5 \]