Composite Function Range Analysis

00:00 5 marks

Consider the composite function \( f(g(x)) \) where \( f(x) = \sqrt{x-1} \) and \( g(x) = 2x+3 \).

(a) Compute the range intervals of \( f(g(x)) \) by first determining the conditions under which \( f(x) \) is defined when applied to \( g(x) \).

(b) Identify and explain the constraints on \( x \) that ensure the composite function \( f(g(x)) \) is defined.

Part a

6 minutes 3 marks

Question

Given that \( f(x) = \sqrt{x-1} \) and \( g(x) = 2x+3 \), compute the range of the composite function \( f(g(x)) \). Begin by determining the condition imposed by \( f(x) \) (i.e. its domain restriction) on the output of \( g(x) \).

Hint Hint

Hint

Ensure that the input for \( f(x) \) is non-negative by setting \( g(x)-1 \geq 0 \).

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Answer Answer

Complete Answer

The composite function simplifies to \( f(g(x)) = \sqrt{2x+2} \) with the condition \( x \geq -1 \). Therefore, its range is \([0, \infty)\).

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Part b

4 minutes 2 marks

Question

Explain the necessary constraints on \( x \) that ensure the composite function \( f(g(x)) \) is defined.

Hint Hint

Hint

Set the expression inside the square root in \( f(x) \) greater than or equal to zero.

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Complete Answer

The constraint is \( x \geq -1 \), since this ensures that \( 2x+2 \geq 0 \) and hence \( f(g(x)) \) is defined.

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