Function Composition and Inverse

00:00 14 marks

Let \(f(x)=2x-1\) and \(g(x)=\frac{1}{x}\) for \(x
eq 0\).

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

(b) Find \((g \circ f)(x)\). State the domain of \((g \circ f)\).

(d) Solve the equation \(f^{-1}(x)=(g \circ f)(x)\).

Part a

3 minutes 2 marks

Question

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

Hint Hint

Hint

Substitute \(g(x)=\frac{1}{x}\) into \(f(x)=2x-1\) to get \(f(g(x))\).

Answer Answer
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Complete Answer

\(f \circ g)(x)=\frac{2}{x}-1\)

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Part b

3 minutes 3 marks

Question

Find \((g \circ f)(x)\). State the domain of \((g \circ f)\).

Hint Hint

Hint

Substitute \(f(x)=2x-1\) into \(g(x)=\frac{1}{x}\) and determine when the denominator is non-zero.

Answer Answer
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Explanation Explanation

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Complete Answer

\((g \circ f)(x)=\frac{1}{2x-1}\) with domain \(\{x \in \mathbb{R} : x
eq \frac{1}{2}\}\).

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Part c

3 minutes 3 marks

Question

Find \(f^{-1}(x)\).

Hint Hint

Hint

Let \(y=2x-1\), swap x and y and then solve for y.

Answer Answer
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Complete Answer

\(f^{-1}(x)=\frac{x+1}{2}\)

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Part d

6 minutes 6 marks

Question

Solve the equation \(f^{-1}(x)=(g \circ f)(x)\).

Hint Hint

Hint

Substitute \(f^{-1}(x)=\frac{x+1}{2}\) and \((g \circ f)(x)=\frac{1}{2x-1}\) into the equation, cross-multiply, and solve the resulting quadratic.

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Complete Answer

The solutions are \(x=-\frac{3}{2}\) or \(x=1\).

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