QQuestionMathematics
QuestionMathematics
Which function is positive for the entire interval (- 3, - 2]?
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Step 1:I'll solve this step by step, carefully examining the given functions:
Step 2:: Identify the interval
- The interval is (- 3, - 2], which means all numbers greater than - 3 and less than or equal to - 2
Step 3:: Evaluate each function at the interval endpoints and within the interval
• At $$x = -2$$: $$f(-2) = (-2)^{2} - 4 = 4 - 4 = 0$$ (zero)
• At x = - 3: f(- 3) = (- 3)^{2} - 4 = 9 - 4 = 5 (positive)
Step 4:: Check function 1's sign
- $$f(x) = x^{2} - 4$$ is always $$\geq 0$$ because it's a parabola that opens upward
- This function is NOT positive for the entire interval
Step 5:: Evaluate Function 2: g(x) = x + 2
- At $$x = -3$$: $$g(-3) = -3 + 2 = -1$$ (negative)
- At x = - 2: g(- 2) = - 2 + 2 = 0 (zero)
Step 6:: Evaluate Function 3: h(x) = -x - 1
- At $$x = -3$$: $$h(-3) = -(-3) - 1 = 3 - 1 = 2$$ (positive)
- At x = - 2: h(- 2) = -(- 2) - 1 = 2 - 1 = 1 (positive)
Step 7:: Determine sign within interval
- For $$h(x) = -x - 1$$, substitute a point between -3 and -2, say -2.5:
h(- 2.5) = -(- 2.5) - 1 = 2.5 - 1 = 1.5 (positive)
Final Answer
h(x) = -x - 1 is the only function positive for the entire interval (- 3, - 2].
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