Intersection Analysis in Piecewise Linear Models

00:00 7 marks

Consider two piecewise linear functions defined as follows:

For \(f(x)\):
\( f(x) = \begin{cases} 2x+3 &\text{if } x \leq 1 \\ -x+5 &\text{if } x > 1 \end{cases} \)

For \(g(x)\) (SL version):
\( g(x) = \begin{cases} x-1 &\text{if } x < 2 \\ 3x-4 &\text{if } x \geq 2 \end{cases} \)

(a) Using the SL definitions, find the intersection point(s) of \(f(x)\) and \(g(x)\) by considering the appropriate domain restrictions.

(b) For an HL extension, suppose the domain of \(g(x)\) is extended to all real numbers by defining \(g(x)=x-1\) for all \(x\). Determine the new intersection point(s) with \(f(x)\).

Part a

5 minutes 3 marks

Question

Solve for the intersection point(s) of \( f(x) = \begin{cases} 2x+3 &\text{if } x \leq 1 \\ -x+5 &\text{if } x > 1 \end{cases} \) and \( g(x) = \begin{cases} x-1 &\text{if } x < 2 \\ 3x-4 &\text{if } x \geq 2 \end{cases} \) considering the given domains.

Hint Hint

Hint

Consider the intervals where both \(f(x)\) and \(g(x)\) are defined. Solve the system of equations for each overlapping interval.

Answer Answer
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Explanation Explanation
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Answer Answer

Complete Answer

The functions intersect at \((-4, -5)\) and \((\frac{9}{4}, \frac{11}{4})\).

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Part b

7 minutes 4 marks

Question

Assuming the HL extension where \(g(x)\) is redefined as \(g(x)=x-1\) for all \(x\), determine the intersection point(s) between \(f(x)\) and this extended \(g(x)\).

Hint Hint

Hint

Use \(g(x) = x-1\) for all real numbers and solve for intersections with each piece of \(f(x)\) separately, considering the domain of each piece of \(f(x)\).

Answer Answer
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Explanation Explanation
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Answer Answer

Complete Answer

The intersections occur at \((-4, -5)\) and \((3, 2)\).

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Answer Explanation Answer Explanation

Formula Booklet