Composite Function Range Analysis

00:00 5 marks

Consider the functions \(f(x)= \frac{1}{x-1}\) and \(g(x)= \sqrt{x+3}\). Their composite is \(f(g(x))= \frac{1}{\sqrt{x+3}-1}\).

Part a

4 minutes 3 marks

Question

Compute the range of the composite function \(f(g(x))= \frac{1}{\sqrt{x+3}-1}\). Introduce the substitution \(t=\sqrt{x+3}\) and analyze the resulting expression \(\frac{1}{t-1}\) by considering the two cases: when \(0 \leq t 1\).

Hint Hint

Hint

Separate the analysis into two cases: one where \(t-1\) is negative and one where it is positive.

Answer Answer
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Answer Answer

Complete Answer

The range of \(f(g(x))\) is \(( -\infty, -1] \cup (0, \infty)\).

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Part b

3 minutes 2 marks

Question

Identify all constraints on \(x\) so that the composite function \(f(g(x))= \frac{1}{\sqrt{x+3}-1}\) is defined.

Hint Hint

Hint

Consider both the domain of \(g(x)=\sqrt{x+3}\) and the condition that \(\sqrt{x+3}\) must not equal 1 for \(f(x)\).

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Complete Answer

\(x\) must satisfy \(x \geq -3\) and \(x
eq -2\) (since \(\sqrt{x+3}=1\) when \(x=-2\)).

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