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NSWMaths AdvancedQuick questions

Year 12: Functions

Quick questions on Composite and inverse functions: existence, formulas, domains and graphs

6short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What are composite functions?
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The composite f∘gf \circ g is the function (f∘g)(x)=f(g(x))(f \circ g)(x) = f(g(x)): apply gg first, then ff to the result.
What is finding an inverse algebraically?
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The inverse f−1f^{-1} "undoes" ff: f−1(f(x))=xf^{-1}(f(x)) = x for x∈dom(f)x \in \text{dom}(f) and f(f−1(y))=yf(f^{-1}(y)) = y for y∈range(f)y \in \text{range}(f).
What is the inverse graph?
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The graph of y=f−1(x)y = f^{-1}(x) is the reflection of the graph of y=f(x)y = f(x) in the line y=xy = x. Point (a,b)(a, b) on ff corresponds to (b,a)(b, a) on f−1f^{-1}. Horizontal asymptotes of ff become vertical asymptotes of f−1f^{-1} and vice versa.
What is see the inverse appear as a reflection, stage by stage?
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Take f(x)=2x−3f(x) = 2x - 3 (whose inverse is f−1(x)=x+32f^{-1}(x) = \frac{x + 3}{2}, found below). The four panels build the inverse purely by reflecting in y=xy = x, with no algebra, so you can see why swapping coordinates is the same as mirroring.
What is restricting the domain?
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For a non-one-to-one function ff, choose a domain on which ff is one-to-one, then invert. Different restrictions give different inverses.
What is wrong domain for the inverse?
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The domain of f−1f^{-1} equals the range of ff. For f(x)=exf(x) = e^x with range (0,∞)(0, \infty), the inverse ln⁡x\ln x has domain (0,∞)(0, \infty).

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