Inverse of composite functions

b. Let g(x) = x² + 1 and h(x) = √x. Consider the composite function ƒ(x) = g(h(x)). Find ƒ⁻¹ directly and then express it in terms of g⁻¹ and h⁻¹

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Start by finding the composite function $ f(x) = g(h(x)) $. Substitute $ h(x) = \sqrt{x} $ into $ g(x) = x^2 + 1 $ to get $ f(x) = g(\sqrt{x}) = (\sqrt{x})^2 + 1 = x + 1 $.

To find the inverse $ f^{-1}(x) $, start with the equation $ y = x + 1 $ and solve for $ x $.

Rearrange the equation $ y = x + 1 $ to express $ x $ in terms of $ y $: $ x = y - 1 $.

Thus, the inverse function is $ f^{-1}(x) = x - 1 $.

Now, express $ f^{-1} $ in terms of $ g^{-1} $ and $ h^{-1} $. Since $ f(x) = g(h(x)) $, we have $ f^{-1}(x) = h^{-1}(g^{-1}(x)) $. Given $ g(x) = x^2 + 1 $, $ g^{-1}(x) = \sqrt{x - 1} $, and $ h(x) = \sqrt{x} $, $ h^{-1}(x) = x^2 $. Therefore, $ f^{-1}(x) = h^{-1}(g^{-1}(x)) = (\sqrt{x - 1})^2 = x - 1 $.

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Composite Functions

A composite function is formed when one function is applied to the result of another function. In this case, if we have functions g(x) and h(x), the composite function f(x) = g(h(x)) means we first apply h to x and then apply g to the result of h. Understanding how to manipulate and analyze composite functions is crucial for finding their inverses.

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An inverse function essentially reverses the effect of the original function. If f(x) takes an input x and produces an output y, then the inverse function f⁻¹(y) takes y back to x. To find the inverse of a composite function, we often need to find the inverses of the individual functions involved and apply them in the reverse order.

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Understanding function notation and operations is essential for working with functions and their inverses. This includes knowing how to denote functions, apply them, and manipulate their outputs. In the context of the question, expressing the inverse of the composite function in terms of the inverses of g and h requires a solid grasp of how to combine and rearrange these functions correctly.

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Domanda del libro di testo

A culture of bacteria has a population of $150$ cells when it is first observed. The population doubles every $12~$\text{hr}$$ , which means its population is governed by the function $p$\left$(t$\right$)=150$\cdot{2^{\frac{t}{12}$}}$ , where $t$ is the number of hours after the first observation.