Composite Function and Intersection Problem

00:00 7 marks

Let \(f(x)=e^{2x}\) and \(g(x)=5x-3\).

(a) Find \((f \circ g)(x)\).
(b) Solve the equation \((f \circ g)(x)=10\).
(c) Find the coordinates of the point where the graph of \(y=f(x)\) intersects the graph of \(y=g^{-1}(x)\).

Part a

3 minutes 2 marks

Question

Find \((f \circ g)(x)\).

Hint Hint

Hint

Substitute \(g(x)=5x-3\) into \(f(x)=e^{2x}\) to get \(f(g(x))=e^{2(5x-3)}\).

Answer Answer
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Complete Answer

\(e^{10x-6}\)

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Part b

3 minutes 2 marks

Question

Solve the equation \((f \circ g)(x)=10\).

Hint Hint

Hint

Set \(e^{10x-6}=10\) and take natural logarithms to solve for x.

Answer Answer
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Complete Answer

\(x = \frac{\ln(10)+6}{10} \approx 0.830\)

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Part c

4 minutes 3 marks

Question

Find the coordinates of the point where the graph of \(y=f(x)\) intersects the graph of \(y=g^{-1}(x)\).

Hint Hint

Hint

First, find the inverse function of g(x) by solving \(y=5x-3\) for x, then set \(f(x)=e^{2x}\) equal to \(g^{-1}(x)\) and solve for x.

Answer Answer
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Complete Answer

The intersection point is approximately \((-1.48, 0.0518)\).

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