Composite Function Transformations Analysis

00:00 9 marks

This question explores function transformations, a key concept in mathematical modeling and computer applications. You will analyze a basic function and apply a sequence of transformations, detailing how each step affects its properties.

Part a

5 minutes 3 marks

Question

Analyze the original function \( f(x) = \sqrt{x} \). Explain its key characteristics including the domain and range, and identify how it is derived from the parent function \( g(x)=x \).

Hint Hint

Hint

Think about the values \(x\) can take for \( \sqrt{x} \) to be defined in real numbers, and the set of possible output values.

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Complete Answer

The function \( f(x) = \sqrt{x} \) has domain \( x \ge 0 \) and range \( y \ge 0 \). It is derived from the linear function \( g(x)=x \) by restricting the domain of \(g(x)\) to non-negative values and then applying the square root operation to the output.

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Part b

5 minutes 3 marks

Question

Apply a horizontal shift of \(-3\) units and a vertical stretch by a factor of 2 to \( f(x)=\sqrt{x} \). Determine the equation for the transformed function \( h(x) \).

Hint Hint

Hint

Remember the order of transformations: horizontal shifts affect the input \(x\) and vertical stretches affect the output \(f(x)\).

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Complete Answer

\( h(x)=2\sqrt{x+3} \)

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Part c

5 minutes 3 marks

Question

Determine the domain and range of the transformed function \( h(x)=2\sqrt{x+3} \). Explain how these have changed from the original function \( f(x) \) due to the applied transformations.

Hint Hint

Hint

Consider how the horizontal shift affects the domain and how the vertical stretch affects the range. Start from the domain and range of \( f(x) = \sqrt{x} \).

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Complete Answer

Domain of \( h(x) \): \( x \geq -3 \); Range of \( h(x) \): \( y \geq 0 \)

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