Function Composition and Inverse Functions

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

Part a

4 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 \(f\Big(\frac{1}{x}\Big)\).

Answer Answer
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Complete Answer

\(\frac{2}{x} – 1\)

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Part b

4 minutes 3 marks

Question

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

Hint Hint

Hint

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

Answer Answer
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Complete Answer

\(\frac{1}{2x-1}\) with \(x
eq \frac{1}{2}\)

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Part c

3 minutes 3 marks

Question

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

Hint Hint

Hint

Let \(y=f(x)\), swap \(x\) and \(y\), and solve for \(y\).

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

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Part d

4 minutes 6 marks

Question

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

Hint Hint

Hint

Set \(\frac{x+1}{2} = \frac{1}{2x-1}\), cross multiply, and solve the resulting quadratic.

Answer Answer
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Complete Answer

The equation simplifies to \(2x^2 + x – 3 = 0\), yielding \(x = -\frac{3}{2}\) or \(x = 1\).

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