Composite Exponential-Logarithmic Function Analysis

00:00 5 marks

Consider the functions defined by \(f(x)=4^{x-2}\) and \(g(x)=\log_4(3x+1)\).

(a) Evaluate the composite function \(f(g(x))\) and simplify your answer.

(b) Determine and simplify the domain and range of the composite function \(f(g(x))\).

Part a

5 minutes 3 marks

Question

Given \(f(x)=4^{x-2}\) and \(g(x)=\log_4(3x+1)\), evaluate the composite function \(f(g(x))\). Provide your answer in its simplest form.

Hint Hint

Hint

Recall the property \(4^{\log_4(b)} = b\) to simplify the expression.

Answer Answer
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Complete Answer

f(g(x)) = \(\frac{3x+1}{16}\)

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Part b

3 minutes 2 marks

Question

Determine and simplify the domain and range of the composite function \(f(g(x))\).

Hint Hint

Hint

Consider the domain of \(g(x)=\log_4(3x+1)\) which requires \(3x+1>0\), and note that the resulting function is linear and consider its possible output values based on the domain.

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Complete Answer

Domain: \(x > -\frac{1}{3}\); Range: \(y > 0\)

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