Piecewise Linear Graph Analysis

00:00 5 marks

The graph of a piecewise function consists of two linear segments. The first segment is defined for \(x < 2\) by the equation \(y = 3x + 1\), and the second segment is defined for \(x \geq 2\) by the equation \(y = -2x + 7\).

(a) Evaluate the function at \(x = 1\), \(x = 2\), and \(x = 3\), and describe the behavior of the function at the breakpoint \(x = 2\).

(b) Determine the domain and range of the function based on these definitions.

Part a

5 minutes 3 marks

Question

Evaluate the function for the following values:

– For \(x = 1\), since \(1 < 2\), use \(y = 3x + 1\).
– For \(x = 2\), use the second segment \(y = -2x + 7\) as \(x \geq 2\).
– For \(x = 3\), again use \(y = -2x + 7\).

Also, explain what happens at \(x = 2\) in terms of continuity.

Hint Hint

Hint

Check each interval condition carefully and compare the value computed at \(x=2\) with the limit as \(x\) approaches 2 from the left.

Answer Answer
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Answer Answer

Complete Answer

For \(x = 1\): \(3(1)+1 = 4\); for \(x = 2\): \(-2(2)+7 = 3\); for \(x = 3\): \(-2(3)+7 = 1\). There is a jump discontinuity at \(x = 2\) because the left-hand limit is \(3(2)+1 = 7\) while the function value at \(x = 2\) is 3.

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Part b

3 minutes 2 marks

Question

Determine the domain and range of the piecewise function defined by:

\(f(x) = 3x+1\) for \(x < 2\) and \(f(x) = -2x+7\) for \(x \geq 2\).

Hint Hint

Hint

Consider the set of all \(x\) values for the domain and the output values from each segment for the range.

Answer Answer
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Explanation Explanation
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Answer Answer

Complete Answer

Domain: all real numbers; Range: \((-\infty, 7)\)

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