Transforming a Piecewise Function

00:00 7 marks

Consider the piecewise-defined function \( f(x) = \begin{cases} x^2 & \text{if } x < 0 \\ \sqrt{x} & \text{if } x \ge 0 \end{cases} \).

Apply the following transformations to obtain a new function \( g(x) \):
– A horizontal shift of 3 units to the right, and
– A vertical stretch by a factor of 2.

(a) Write the expression for \( g(x) \) for each segment.

(b) Determine whether \( g(x) \) is continuous at the point where the definition changes.

Part a

6 minutes 4 marks

Question

Transform each piece of \( f(x) \) to obtain \( g(x) \) by applying the horizontal shift and vertical stretch.

Provide the transformed function expressions for:
– For \( x < 0 \): starting from \( f(x) = x^2 \)
– For \( x \ge 0 \): starting from \( f(x) = \sqrt{x} \)

Hint Hint

Hint

Replace \( x \) with \( x-3 \) in each piece and multiply the function output by 2. Note the domain shift due to the horizontal translation.

Answer Answer
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Answer Answer

Complete Answer

For \( x < 0 \): \( g(x) = 2(x-3)^2 \) with the domain adjusted to \( x < 3 \); for \( x \ge 0 \): \( g(x) = 2\sqrt{x-3} \) with the domain adjusted to \( x \ge 3 \).

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Part b

4 minutes 3 marks

Question

Determine whether the transformed function \( g(x) \) is continuous at the junction point \( x = 3 \). Provide justification for your answer.

Hint Hint

Hint

Compute the left-hand and right-hand limits of \( g(x) \) as \( x \) approaches 3 and compare them with \( g(3) \).

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Complete Answer

The function \( g(x) \) is continuous at \( x = 3 \) because both the left-hand limit and the right-hand limit equal \( g(3) = 0 \).

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