Use compositions to answer each of the following: (a) If...

Use compositions to answer each of the following:
(a) If $g(x)=x^{2}$ and $f(g(x))=\frac{1}{x^{2}+1}$, what is $f(x)$ ?
(b) If $h(x)=x^{2}-1$ and $h(l(x))=\frac{1}{x^{4}}-1$, what is $l(x)$ ?
(c) If $u(x)=\frac{1}{1-x}$ and $y(u(x))=\frac{1}{1-x}$, what is $y(x)$ ?

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Key Concepts

Solving for Unknown Functions

Solving for unknown functions using compositions requires setting up equations from the relationships provided and then isolating the unknown function. This typically involves expressing the composite function in terms of the inner function’s outputs, equating that expression to a given target, and thereby deducing the form of the unknown function.

Function Composition

Function composition is the process of applying one function to the result of another. In mathematical terms, if you have two functions f and g, the composite function f(g(x)) involves computing g(x) first and then applying f to that result. This concept is fundamental in analyzing how functions interact with one another.

Substitution Method in Functional Equations

The substitution method involves replacing a variable or an expression with another equivalent expression. When dealing with function compositions, this method is used to substitute the output of an inner function into the outer function’s formula, enabling the determination of unknown functions by matching the given composite expression.

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