Composite Function

Let a function f be f:x\rightarrow x-3 and a function g such that g \circ f : x \rightarrow 4x^2 - 20x + 25. What is the function g(x) ?

Solution :

To tackle problems involving composite functions, we have to clear other functions by using the property

f\circ f^{-1} : x \rightarrow x

\begin{aligned}  g \circ f &: x \rightarrow 4x^2 - 20x + 25 \\ g \circ f &: x \rightarrow (2x-5)^2 \\ g \circ f \circ f^{-1} &: x \rightarrow \big [2f^{-1}(x) - 5\big ]^2  \end{aligned}

It’s clear that f^{-1}(x) = x + 3. Substitute into the expression we have

g:x\rightarrow (2x+1)^2

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Composite Function

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