Given f(x)=√(2-4 x) and g(x)=-(3)/(x), find the following: a. (g ∘f)(x) b. the domain of (g ∘f)(

step 1 So here I have two functions and I want to find G composed of f at x. And I'll start by doing G

Given f(x)=√(2-4 x) and g(x)=-(3)/(x), find the following:...

Key Concepts

Function Composition

Function composition involves applying one function to the result of another. In the context of (g ? f)(x), it means that you first apply f to x and then apply g to the outcome of f(x). Understanding this concept is crucial as it simplifies the evaluation of combined functions and highlights the order in which functions are applied.

Domain of a Composite Function

The domain of a composite function is determined by considering the restrictions not only from the inner function but also from the outer function applied to the output of the inner function. This means you must include only those x-values for which the inner function is defined and for which the output of the inner function also lies in the domain of the outer function.

Square Root Function Domain Restrictions

The square root function is defined only when its radicand is non-negative. Therefore, when dealing with functions that involve a square root, it is important to set the expression inside the square root greater than or equal to zero and solve for x, ensuring that the function outputs real numbers.

Rational Function Domain Restrictions

Rational functions, which involve division by a variable expression, are undefined when the denominator is zero. When composing functions that include a rational function, it is necessary to identify any values of the variable that would make the denominator zero, and these values must be excluded from the overall domain.

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