MediumMathematicsHigh School (11–12)Functions and Calculus › Composite and Inverse FunctionsConceptual

Let f(x) = x - 4 and g(x) = 2x. Find (g ° f)⁻¹, and check that it equals f⁻¹ ° g⁻¹ but not g⁻¹ °…

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

Let $f(x) = x - 4$ and $g(x) = 2x$ . Find $(g \circ f)^{-1}$ , and check that it equals $f^{-1} \circ g^{-1}$ but not $g^{-1} \circ f^{-1}$ .

Need a reminder? The key ideaShow

$f$ has an inverse exactly when it is one-one and onto. To find $f^{-1}$ , solve $y = f(x)$ for $x$ . Its graph is the reflection of $f$ in $y = x$ , and $(g \circ f)^{-1} = f^{-1} \circ g^{-1}$ .

The worked solution

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Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

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Let f(x) = x - 4 and g(x) = 2x. Find (g ° f)⁻¹, and check that it equals f⁻¹ ° g⁻¹ but not g⁻¹ °…

The worked solution

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