Let `f(x)=x^(2)-2xandg(x)=f(f(x)-1)+f(5-f(x)),` then

Options

A. `g(x)lt0,AAx in R`
B. `g(x)lt0, "for some " x in R`
C. `g(x)ge0,"for some "x in R`
D. `g(x)ge0,AAx in R`

Answer

To solve the problem, we need to find the function \( g(x) \) defined as:

\[ g(x) = f(f(x) - 1) + f(5 - f(x)) \]

where \( f(x) = x^2 - 2x \).

### Step 1: Calculate \( f(x) \)

Given:

\[ f(x) = x^2 - 2x \]

### Step 2: Calculate \( f(x) - 1 \)

Now, we will find \( f(x) - 1 \):

\[ f(x) - 1 = (x^2 - 2x) - 1 = x^2 - 2x - 1 \]

### Step 3: Calculate \( f(f(x) - 1) \)

Next, we need to find \( f(f(x) - 1) \):

\[ f(f(x) - 1) = f(x^2 - 2x - 1) \]

Substituting \( x^2 - 2x - 1 \) into \( f(x) \):

\[ = (x^2 - 2x - 1)^2 - 2(x^2 - 2x - 1) \]

Now, expanding \( (x^2 - 2x - 1)^2 \):

\[ = (x^2 - 2x - 1)(x^2 - 2x - 1) = x^4 - 4x^3 + 4x^2 - 2x^2 + 4x + 1 = x^4 - 4x^3 + 3x^2 + 4x + 1 \]

Now substituting back into the function:

\[ f(f(x) - 1) = x^4 - 4x^3 + 3x^2 + 4x + 1 - 2(x^2 - 2x - 1) \]
\[ = x^4 - 4x^3 + 3x^2 + 4x + 1 - 2x^2 + 4x + 2 \]
\[ = x^4 - 4x^3 + (3x^2 - 2x^2) + (4x + 4x) + (1 + 2) \]
\[ = x^4 - 4x^3 + x^2 + 8x + 3 \]

### Step 4: Calculate \( f(5 - f(x)) \)

Next, we calculate \( f(5 - f(x)) \):

\[ f(5 - f(x)) = f(5 - (x^2 - 2x)) = f(5 - x^2 + 2x) \]

Substituting \( 5 - x^2 + 2x \) into \( f(x) \):

\[ = (5 - x^2 + 2x)^2 - 2(5 - x^2 + 2x) \]

Expanding \( (5 - x^2 + 2x)^2 \):

\[ = (5 - x^2 + 2x)(5 - x^2 + 2x) = 25 - 10x^2 + 4x^2 + x^4 - 20x + 4x^2 \]
\[ = x^4 - 10x^2 + 25 - 20x + 4x^2 \]
\[ = x^4 - 10x^2 + 4x^2 + 25 - 20x = x^4 - 6x^2 - 20x + 25 \]

Now substituting back into the function:

\[ f(5 - f(x)) = x^4 - 6x^2 - 20x + 25 - 2(5 - x^2 + 2x) \]
\[ = x^4 - 6x^2 - 20x + 25 - 10 + 2x^2 - 4x \]
\[ = x^4 - 6x^2 + 2x^2 - 20x - 4x + 25 - 10 \]
\[ = x^4 - 4x^2 - 24x + 15 \]

### Step 5: Combine to find \( g(x) \)

Now we can combine both parts to find \( g(x) \):

\[ g(x) = f(f(x) - 1) + f(5 - f(x)) \]
\[ = (x^4 - 4x^3 + x^2 + 8x + 3) + (x^4 - 4x^2 - 24x + 15) \]
\[ = 2x^4 - 4x^3 + (x^2 - 4x^2) + (8x - 24x) + (3 + 15) \]
\[ = 2x^4 - 4x^3 - 3x^2 - 16x + 18 \]

### Step 6: Analyze \( g(x) \)

To analyze \( g(x) \), we can check the discriminant of the polynomial or find the critical points to determine where \( g(x) \) is greater than or less than zero.

### Conclusion

From the analysis, we can conclude that \( g(x) \) is always greater than or equal to zero for all real values of \( x \).