Let `f(x)=x^2-2x-1 AA xinR` Let `f:(-oo, a]->[b, oo)`, where a is the largest real number for which f(x) is bijective. If `f : R->R` , `g(x) = f(x) + 3x-1` , then the least value of function `y = g(|x|)` is

Options

A. `(-2,-1)`
B. `(-2,0)`
C. `(-1,0)`
D. `(0,1)`

Explanation

To solve the given problem step by step, we will follow the outlined process:

### Step 1: Define the function f(x)
The function is given as:
\[ f(x) = x^2 - 2x - 1 \]

### Step 2: Determine the bijective nature of f(x)
To find the largest real number \( a \) for which \( f(x) \) is bijective, we need to find the vertex of the quadratic function. The vertex form of a quadratic \( ax^2 + bx + c \) can be found using:
\[ x = -\frac{b}{2a} \]
Here, \( a = 1 \) and \( b = -2 \):
\[ a = -\frac{-2}{2 \cdot 1} = 1 \]

Thus, the largest real number \( a \) is 1.

### Step 3: Find the range of f(x) when x is in the domain (-∞, 1]
To find the minimum value of \( f(x) \) at \( x = 1 \):
\[ f(1) = 1^2 - 2 \cdot 1 - 1 = 1 - 2 - 1 = -2 \]

Thus, the range of \( f(x) \) is:
\[ f(x) : (-\infty, -2] \]

### Step 4: Define the function g(x)
The function \( g(x) \) is defined as:
\[ g(x) = f(x) + 3x - 1 \]
Substituting \( f(x) \):
\[ g(x) = (x^2 - 2x - 1) + 3x - 1 \]
Combine like terms:
\[ g(x) = x^2 + x - 2 \]

### Step 5: Complete the square for g(x)
To find the minimum value, we will complete the square:
\[ g(x) = x^2 + x - 2 \]
Taking half of the coefficient of \( x \) (which is \( \frac{1}{2} \)), we square it:
\[ \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]
Now, we can rewrite \( g(x) \):
\[ g(x) = \left(x + \frac{1}{2}\right)^2 - \frac{1}{4} - 2 \]
\[ g(x) = \left(x + \frac{1}{2}\right)^2 - \frac{1}{4} - \frac{8}{4} \]
\[ g(x) = \left(x + \frac{1}{2}\right)^2 - \frac{9}{4} \]

### Step 6: Find the least value of g(|x|)
Now we need to find \( g(|x|) \):
\[ g(|x|) = \left(|x| + \frac{1}{2}\right)^2 - \frac{9}{4} \]

Since \( |x| \) is always non-negative, the minimum value of \( |x| \) is 0:
\[ g(0) = \left(0 + \frac{1}{2}\right)^2 - \frac{9}{4} \]
\[ g(0) = \frac{1}{4} - \frac{9}{4} = -\frac{8}{4} = -2 \]

### Conclusion
The least value of the function \( y = g(|x|) \) is:
\[ \boxed{-2} \]