Consider the piecewise-defined function:
For \(x < 2\): \(f(x) = m*x + n\)
For \(x \geq 2\): \(f(x) = p*x^2 + q\)
Ensure that the function \(f(x)\) is continuous at \(x = 2\).
Follow these steps:
(a) Define the intervals for each piece clearly.
(b) Derive an equation relating \(m\), \(n\), \(p\), and \(q\) that guarantees continuity at \(x = 2\).
(c) Given that \(m = 1\), \(p = 0.5\), and \(f(2) = 3\), determine the values of \(n\) and \(q\) so that \(f\) is continuous at \(x = 2\).
Question
Define the intervals for which each expression of \(f(x)\) is valid.
Hint
Hint
Hint
Consider the definition of the piecewise function and identify the range of \(x\)-values for each piece based on the boundary condition at \(x = 2\).
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- Formula Booklet
Complete Answer
For \(x < 2\): the interval is \((-\infty, 2)\). For \(x \geq 2\): the interval is \([2, \infty)\).
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Question
Derive the condition relating \(m\), \(n\), \(p\), and \(q\) that ensures continuity of \(f\) at \(x = 2\).
Hint
Hint
Hint
For \(f(x)\) to be continuous at \(x = 2\), the left-hand limit and the right-hand limit at \(x = 2\) must be equal. Set the expressions for \(f(x)\) as \(x\) approaches 2 from both sides equal to each other.
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- Formula Booklet
Complete Answer
For continuity at \(x = 2\), we require the left-hand limit to equal the right-hand limit. This means: \(m*(2) + n = p*(2)^2 + q\), which simplifies to \(2m + n = 4p + q\).
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Question
Given that \(m = 1\), \(p = 0.5\), and \(f(2) = 3\), determine the values of \(n\) and \(q\) so that \(f\) is continuous at \(x = 2\).
Hint
Hint
Hint
Substitute the given values of \(m\) and \(p\) into the continuity condition derived in part (b). Then use the condition \(f(2) = 3\) along with the definition of \(f(x)\) for \(x \geq 2\) to solve for \(n\) and \(q\).
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- Formula Booklet
Complete Answer
Substituting \(m = 1\) and \(p = 0.5\) into \(2m + n = 4p + q\) gives \(2(1) + n = 4(0.5) + q\), which simplifies to \(2 + n = 2 + q\), hence \(n = q\). Since \(f(2) = 3\) and for \(x \geq 2\), \(f(x) = p*x^2 + q\), we have \(f(2) = 0.5*(2)^2 + q = 2 + q = 3\). Solving for \(q\) gives \(q = 1\). Since \(n = q\), \(n = 1\).
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