Question

the graph of the function ( f(x) = -(x + 3)(x - 1) ) is shown below.
which statement about the function is true?

• the function is positive for all real values of ( x ) where ( x < -1 ).
• the function is negative for all real values of ( x ) where ( x < -3 ) and where ( x > 1 ).
• the function is positive for all real values of ( x ) where ( x > 0 ).
• the function is negative for all real values of ( x ) where ( x < -3 ) or ( x > -1 ).

Explanation:

Step1: Find the roots of the function

The function is \( f(x) = -(x + 3)(x - 1) \). To find the roots, set \( f(x)=0 \):
\( -(x + 3)(x - 1)=0 \) implies \( x + 3 = 0 \) or \( x - 1 = 0 \), so the roots are \( x=-3 \) and \( x = 1 \).

Step2: Analyze the parabola's direction

The coefficient of \( x^2 \) (when expanded) is \( -1 \) (since \( -(x + 3)(x - 1)=-(x^2 - x + 3x - 3)=-x^2 - 2x + 3 \)), so the parabola opens downward (because the coefficient of \( x^2 \) is negative).

Step3: Determine where the function is positive/negative

For a downward - opening parabola, the function is positive between its two roots and negative outside the interval between the roots.
The roots are \( x=-3 \) and \( x = 1 \). So the function is positive when \( -31 \).

Now let's analyze each option:

• Option 1: "The function is positive for all real values of \( x \) where \( x < - 1 \)". This is false because when \( x < - 3 \), the function is negative.
• Option 2: "The function is negative for all real values of \( x \) where \( x < - 3 \) and where \( x>1 \)". This is true as we found that the function is negative when \( x < - 3 \) or \( x>1 \) (the "and" here can be considered as the union of the two intervals in the context of the option's wording).
• Option 3: "The function is positive for all real values of \( x \) where \( x>0 \)". When \( x>1 \), the function is negative, so this is false.
• Option 4: "The function is negative for all real values of \( x \) where \( x < - 3 \) or \( x>-1 \)". When \( - 3

Answer:

The function is negative for all real values of \( x \) where \( x < - 3 \) and where \( x>1 \) (the second option).