Consider the functions `f(x) =sqrtx` and `g(x) x^2-b`. In the standard (x, y) coordinate plane, `y = f(g(x))` passes through the point (3, 5). What is the value of b ?

Options

A. 16
B. 4
C. `-2`
D. `-16`

Answer

To solve the problem, we need to find the value of \( b \) such that the function \( y = f(g(x)) \) passes through the point \( (3, 5) \). We are given the functions:

- \( f(x) = \sqrt{x} \)
- \( g(x) = x^2 - b \)

### Step 1: Substitute \( g(x) \) into \( f(x) \)

We start by substituting \( g(x) \) into \( f(x) \):

\[ y = f(g(x)) = f(x^2 - b) = \sqrt{x^2 - b} \]

### Step 2: Use the given point \( (3, 5) \)

We know that the function passes through the point \( (3, 5) \). This means when \( x = 3 \), \( y \) should equal 5. We can set up the equation:

\[ 5 = \sqrt{3^2 - b} \]

### Step 3: Square both sides to eliminate the square root

To eliminate the square root, we square both sides of the equation:

\[ 5^2 = 3^2 - b \]

This simplifies to:

\[ 25 = 9 - b \]

### Step 4: Solve for \( b \)

Now, we can solve for \( b \) by isolating it:

\[ b = 9 - 25 \]
\[ b = -16 \]

### Conclusion

The value of \( b \) is \( -16 \).

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