Math  /  Algebra

QuestionLet z(x)=tan(x)z(x)=\sqrt{\tan (x)}. Which of the following best describes its fundamental algebraic structure? composition: A composition f(g(x))f(g(x)) of basic functions sum. A sum f(x)+g(x)f(x)+g(x) of basic functions product. A product f(x)g(x)f(x) \cdot g(x) of basic functions quotient. A quotient f(x)/g(x)f(x) / g(x) of basic functions where f(x)=g(x)=\begin{array}{l} f(x)=\square \\ g(x)=\square \end{array}

Studdy Solution

STEP 1

1. The function z(x)=tan(x) z(x) = \sqrt{\tan(x)} is given.
2. We need to identify the fundamental algebraic structure of this function.
3. The options for the structure are composition, sum, product, or quotient of basic functions.

STEP 2

1. Identify the basic functions involved in z(x) z(x) .
2. Determine the algebraic operation that combines these basic functions.
3. Match the operation to one of the given options: composition, sum, product, or quotient.

STEP 3

Identify the basic functions involved in z(x)=tan(x) z(x) = \sqrt{\tan(x)} .
The function involves: - The square root function, which can be represented as f(x)=x f(x) = \sqrt{x} . - The tangent function, which can be represented as g(x)=tan(x) g(x) = \tan(x) .

STEP 4

Determine the algebraic operation that combines these basic functions.
The function z(x)=tan(x) z(x) = \sqrt{\tan(x)} is formed by applying the square root function to the result of the tangent function. This is a composition of functions.

STEP 5

Match the operation to one of the given options.
The operation is a composition of functions, which corresponds to the option "composition: A composition f(g(x)) f(g(x)) of basic functions".
The fundamental algebraic structure of z(x)=tan(x) z(x) = \sqrt{\tan(x)} is best described as a composition: f(g(x)) f(g(x)) of basic functions, where: f(x)=x f(x) = \sqrt{x} g(x)=tan(x) g(x) = \tan(x)

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