Let `f (x)=x^(2) -2x -3, x ge 1 and g (x)=1 +sqrt(x+4), x ge-4` then the number of real solution os equation `f (x) =g (x)` is/are

Options

A. 0
B. 1
C. 2
D. 4

Answer

To find the number of real solutions of the equation \( f(x) = g(x) \), where \( f(x) = x^2 - 2x - 3 \) for \( x \geq 1 \) and \( g(x) = 1 + \sqrt{x + 4} \) for \( x \geq -4 \), we will follow these steps:

### Step 1: Analyze the function \( f(x) \)

1. **Find the roots of \( f(x) \)**:
\[ f(x) = x^2 - 2x - 3 = 0 \]
Factoring gives:
\[ (x - 3)(x + 1) = 0 \]
Thus, the roots are \( x = 3 \) and \( x = -1 \).

2. **Determine the minimum value of \( f(x) \)**:
The vertex of a quadratic function \( ax^2 + bx + c \) occurs at \( x = -\frac{b}{2a} \). Here, \( a = 1 \) and \( b = -2 \):
\[ x = -\frac{-2}{2 \cdot 1} = 1 \]
Now, calculate \( f(1) \):
\[ f(1) = 1^2 - 2 \cdot 1 - 3 = 1 - 2 - 3 = -4 \]
Thus, the minimum value of \( f(x) \) at \( x = 1 \) is \( -4 \).

3. **Behavior of \( f(x) \)**:
Since \( f(x) \) is a quadratic function that opens upwards, it decreases until \( x = 1 \) and then increases. The function \( f(x) \) is below the x-axis for \( x \) between \( -1 \) and \( 3 \), and above the x-axis for \( x > 3 \).

### Step 2: Analyze the function \( g(x) \)

1. **Find the range of \( g(x) \)**:
\[ g(x) = 1 + \sqrt{x + 4} \]
The minimum value occurs at \( x = -4 \):
\[ g(-4) = 1 + \sqrt{-4 + 4} = 1 + 0 = 1 \]
As \( x \) increases, \( g(x) \) increases without bound. Therefore, the range of \( g(x) \) is \( [1, \infty) \).

2. **Behavior of \( g(x) \)**:
The function \( g(x) \) is always increasing because the derivative \( g'(x) = \frac{1}{2\sqrt{x + 4}} \) is always positive for \( x \geq -4 \).

### Step 3: Find the intersection points \( f(x) = g(x) \)

1. **Set \( f(x) = g(x) \)**:
\[ x^2 - 2x - 3 = 1 + \sqrt{x + 4} \]
Rearranging gives:
\[ x^2 - 2x - 4 = \sqrt{x + 4} \]
Squaring both sides:
\[ (x^2 - 2x - 4)^2 = x + 4 \]
Expanding and simplifying:
\[ x^4 - 4x^3 + 4x^2 - 8x + 16 = x + 4 \]
\[ x^4 - 4x^3 + 4x^2 - 9x + 12 = 0 \]

2. **Finding the number of real roots**:
We can analyze the behavior of the polynomial \( x^4 - 4x^3 + 4x^2 - 9x + 12 \) using numerical methods or graphing. However, since we know the behavior of \( f(x) \) and \( g(x) \):
- \( f(x) \) has a minimum of \( -4 \) at \( x = 1 \).
- \( g(x) \) starts at \( 1 \) and increases.

Since \( f(1) = -4 < 1 \) and \( f(x) \) increases to \( \infty \) as \( x \) increases past \( 3 \), we can conclude that \( f(x) \) will intersect \( g(x) \) at two points (one before \( x = 3 \) and one after).

### Conclusion

The number of real solutions to the equation \( f(x) = g(x) \) is **2**.