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To determine the correct behavior of the function [tex]\( f(x) = -(x+3)(x-1) \)[/tex] over the given intervals, we first analyze its roots and general shape by following these logical steps:
1. Identify the roots of the function:
The function [tex]\( f(x) = -(x+3)(x-1) \)[/tex] can be rewritten as:
[tex]\[ f(x) = -(x^2 - 2x - 3) \][/tex]
Setting [tex]\( f(x) = 0 \)[/tex] gives:
[tex]\[ -(x^2 - 2x - 3) = 0 \][/tex]
[tex]\[ x^2 - 2x - 3 = 0 \][/tex]
Factoring the quadratic equation:
[tex]\[ (x+3)(x-1) = 0 \][/tex]
So, the roots are:
[tex]\[ x = -3 \quad \text{and} \quad x = 1 \][/tex]
2. Determine the behavior of the function at the roots and between the roots:
- At [tex]\( x = -3 \)[/tex] and [tex]\( x = 1 \)[/tex], [tex]\( f(x) = 0 \)[/tex].
- When [tex]\( x < -3 \)[/tex], both [tex]\( (x+3) \)[/tex] and [tex]\( (x-1) \)[/tex] are negative, but the combined product being multiplied by the negative makes [tex]\( f(x) \)[/tex] positive.
- Between [tex]\( x = -3 \)[/tex] and [tex]\( x = 1 \)[/tex] (i.e., [tex]\(-3 < x < 1\)[/tex]), [tex]\( (x+3) \)[/tex] is positive and [tex]\((x-1) \)[/tex] is negative, and multiplying these and then negating gives a positive number, so [tex]\( f(x) \)[/tex] is negative.
- When [tex]\( x > 1 \)[/tex], both [tex]\( (x+3) \)[/tex] and [tex]\( (x-1) \)[/tex] are positive, thus their product is positive, and negating it makes [tex]\( f(x) \)[/tex] negative.
3. Summarize the intervals over which the function [tex]\( f(x) \)[/tex] is positive or negative:
- [tex]\( f(x) < 0 \)[/tex] for [tex]\( -3 < x < 1 \)[/tex].
- [tex]\( f(x) > 0 \)[/tex] for [tex]\( x < -3 \)[/tex] or [tex]\( x > 1 \)[/tex].
4. Match the intervals with the given statements:
- 'The function is positive for all real values of [tex]\( x \)[/tex] where [tex]\( x > 0 \)[/tex]' is incorrect.
- 'The function is negative for all real values of [tex]\( x \)[/tex] where [tex]\( x < -3 \)[/tex] and where [tex]\( x > 1 \)[/tex]' is incorrect.
- 'The function is negative for all real values of [tex]\( x \)[/tex] where [tex]\( x < -3 \)[/tex] or [tex]\( x > -1 \)[/tex]' is incorrect.
- 'The function is positive for all real values of [tex]\( x \)[/tex] where [tex]\( x < -1 \)[/tex]' is the correct statement.
Therefore, the correct statement about the function is:
[tex]\[ \text{The function is positive for all real values of } x \text{ where } x < -1. \][/tex]
1. Identify the roots of the function:
The function [tex]\( f(x) = -(x+3)(x-1) \)[/tex] can be rewritten as:
[tex]\[ f(x) = -(x^2 - 2x - 3) \][/tex]
Setting [tex]\( f(x) = 0 \)[/tex] gives:
[tex]\[ -(x^2 - 2x - 3) = 0 \][/tex]
[tex]\[ x^2 - 2x - 3 = 0 \][/tex]
Factoring the quadratic equation:
[tex]\[ (x+3)(x-1) = 0 \][/tex]
So, the roots are:
[tex]\[ x = -3 \quad \text{and} \quad x = 1 \][/tex]
2. Determine the behavior of the function at the roots and between the roots:
- At [tex]\( x = -3 \)[/tex] and [tex]\( x = 1 \)[/tex], [tex]\( f(x) = 0 \)[/tex].
- When [tex]\( x < -3 \)[/tex], both [tex]\( (x+3) \)[/tex] and [tex]\( (x-1) \)[/tex] are negative, but the combined product being multiplied by the negative makes [tex]\( f(x) \)[/tex] positive.
- Between [tex]\( x = -3 \)[/tex] and [tex]\( x = 1 \)[/tex] (i.e., [tex]\(-3 < x < 1\)[/tex]), [tex]\( (x+3) \)[/tex] is positive and [tex]\((x-1) \)[/tex] is negative, and multiplying these and then negating gives a positive number, so [tex]\( f(x) \)[/tex] is negative.
- When [tex]\( x > 1 \)[/tex], both [tex]\( (x+3) \)[/tex] and [tex]\( (x-1) \)[/tex] are positive, thus their product is positive, and negating it makes [tex]\( f(x) \)[/tex] negative.
3. Summarize the intervals over which the function [tex]\( f(x) \)[/tex] is positive or negative:
- [tex]\( f(x) < 0 \)[/tex] for [tex]\( -3 < x < 1 \)[/tex].
- [tex]\( f(x) > 0 \)[/tex] for [tex]\( x < -3 \)[/tex] or [tex]\( x > 1 \)[/tex].
4. Match the intervals with the given statements:
- 'The function is positive for all real values of [tex]\( x \)[/tex] where [tex]\( x > 0 \)[/tex]' is incorrect.
- 'The function is negative for all real values of [tex]\( x \)[/tex] where [tex]\( x < -3 \)[/tex] and where [tex]\( x > 1 \)[/tex]' is incorrect.
- 'The function is negative for all real values of [tex]\( x \)[/tex] where [tex]\( x < -3 \)[/tex] or [tex]\( x > -1 \)[/tex]' is incorrect.
- 'The function is positive for all real values of [tex]\( x \)[/tex] where [tex]\( x < -1 \)[/tex]' is the correct statement.
Therefore, the correct statement about the function is:
[tex]\[ \text{The function is positive for all real values of } x \text{ where } x < -1. \][/tex]
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