Analyzing a Piecewise Step Function

00:00 5 marks

Consider the step function \(f(x) = \lfloor x \rfloor\) defined for \(1 \le x \le 5\) as a model for pricing tiers. Analyze its behavior in terms of the intervals over which it is constant and the jump discontinuities present.

Part a

4 minutes 3 marks

Question

Evaluate the intervals over which \(f(x) = \lfloor x \rfloor\) is constant for \(1 \le x \le 5\). Identify the x-values where the function exhibits jump discontinuities.

Hint Hint

Hint

Consider the definition of the floor function. For what range of x-values does \(\lfloor x \rfloor\) yield the same integer output?

Answer Answer
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Answer Answer

Complete Answer

The function \(f(x) = \lfloor x \rfloor\) is constant on the intervals \([1,2)\), \([2,3)\), \([3,4)\), and \([4,5)\). Jump discontinuities occur at \(x = 2, 3, 4, 5\).

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Part b

4 minutes 2 marks

Question

Sketch the graph of \(f(x) = \lfloor x \rfloor\) for \(1 \le x \le 5\). On your sketch, clearly indicate the step intervals and jump discontinuities using appropriate graphical conventions (solid dots and open circles).

Hint Hint

Hint

For each interval \([n, n+1)\), the graph is a horizontal line segment at \(y = n\). Remember to use solid dots for included endpoints and open circles for excluded endpoints to represent jump discontinuities.

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Answer Answer

Complete Answer

A sketch showing horizontal line segments at \(y=1\) for \([1,2)\), \(y=2\) for \([2,3)\), \(y=3\) for \([3,4)\), and \(y=4\) for \([4,5)\), each with a solid dot at the left endpoint and an open circle at the right endpoint. At \(x=5\), there should be a solid dot at \((5, 5)\).

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