Definition and Classification of Elementary Functions

Approach:

First state the criterion for an elementary function — one built from basic elementary functions by a finite number of arithmetic and composition operations — then match each of the four functions against that criterion.

Steps:

1. Criterion: power functions, exponential functions, logarithmic functions, trigonometric functions and inverse trigonometric functions are the basic elementary functions; anything obtained from them by a finite number of arithmetic and composition operations is an elementary function.
2. $y = x^2$ is a power function, $y = \sin x$ is a trigonometric function and $y = \ln x$ is a logarithmic function, so all three are basic elementary functions and therefore elementary functions.
3. $y = \Gamma(x)$ is the gamma function, a special function that cannot be obtained from the basic elementary functions by a finite number of arithmetic and composition operations, so it is not an elementary function.

Answer:

$y = x^2$, $y = \sin x$, $y = \ln x$ are elementary functions, $y = \Gamma(x)$ is not (Gamma function is a special function).

Approach:

An algebraic function may only contain algebraic operations (addition, subtraction, multiplication, division and root extraction); as soon as a transcendental operation such as exponentiation, logarithm or trigonometry appears, the function is transcendental instead.

Steps:

1. Algebraic functions allow only algebraic operations (addition, subtraction, multiplication, division, root extraction); a single transcendental operation disqualifies the expression.
2. $y = \sqrt{x}$ contains only root extraction and $y = \frac{x^2 + 1}{x - 1}$ contains only the four arithmetic operations, so both are algebraic functions.
3. $y = e^x$ contains an exponential operation and $y = \sin x$ contains a trigonometric operation, so both are transcendental rather than algebraic functions.

Answer:

$y = \sqrt{x}$, $y = \frac{x^2+1}{x-1}$ are algebraic functions, $y = e^x$, $y = \sin x$ are not.

Approach:

A transcendental function is one whose expression contains a transcendental operation such as exponentiation, logarithm or trigonometry; check each of the four functions for such an operation.

Steps:

1. A transcendental function must contain an exponential, logarithmic or trigonometric operation; together with the algebraic functions these two classes make up the elementary functions.
2. $y = \ln x$ contains a logarithmic operation and $y = \sin x$ contains a trigonometric operation, so both are transcendental functions.
3. $y = x^3$ uses only a power with an integer exponent together with addition, and $y = \sqrt{x}$ uses only root extraction, so both are algebraic functions and hence not transcendental.

Answer:

$y = \ln x$, $y = \sin x$ are transcendental functions, $y = x^3$, $y = \sqrt{x}$ are not.

Approach:

Identify which operations occur in the expression and how the transcendental pieces are combined; that decides the class the function belongs to inside the elementary functions.

Steps:

1. In $y = e^{\sin x}$ the outer layer is the exponential function $e^u$ and the inner layer is the trigonometric function $\sin x$, so the expression has a composite structure.
2. Both the exponential operation and the trigonometric operation are transcendental, and any function containing a transcendental operation is classified as transcendental.
3. Composition is itself one of the ways elementary functions are built, so $e^{\sin x}$ remains an elementary function while belonging to the transcendental class.

Answer:

It belongs to transcendental functions because it contains the composition of exponential and trigonometric functions.

Approach:

Check the expression against the criterion for algebraic functions, listing every operation that appears, and then decide whether it satisfies the transcendental criterion instead.

Steps:

1. The expression $y = \sqrt{x^2 + 1}$ contains addition, squaring and root extraction, all of which are algebraic operations.
2. No exponential, logarithmic or trigonometric operation appears, so the transcendental criterion is not met.
3. A radical can also be written as a power, $(x^2 + 1)^{1/2}$, which is again a composition of a power function with a linear function, so the function is still algebraic.

Answer:

It belongs to algebraic functions because it only involves algebraic operations and root extraction.

Approach:

Compare each option with the criterion for transcendental functions: an expression containing an exponential, logarithmic or trigonometric operation is transcendental.

Steps:

1. (A) $y = x^2$ contains only a power operation, so it is an algebraic function.
2. (B) $y = \sin x$ contains a trigonometric operation, so it is a transcendental function.
3. (C) $y = \ln x$ contains a logarithmic operation, so it is a transcendental function.
4. (D) $y = \sqrt{x}$ contains only root extraction, so it is an algebraic function.

Answer:

(B), (C) are transcendental functions, (A), (D) are not.

Approach:

Inspect every part of the fractional expression to see whether only algebraic operations occur, which decides whether it is an algebraic function or a transcendental one.

Steps:

1. The numerator $x^2 + 1$ contains only a power and an addition, so it is a polynomial and therefore algebraic.
2. The denominator $x - 1$ contains only a subtraction, so it is a linear polynomial and therefore algebraic.
3. Dividing the two keeps the expression free of transcendental operations, so the whole function is a rational function, and a rational function is an algebraic function.

Answer:

Yes, it is an algebraic function because it only involves algebraic operations such as addition, subtraction, multiplication, and division.

Approach:

Start from the operations occurring in the expression to decide which class the function falls into, and then explain why this composite structure still sits inside the elementary functions.

Steps:

1. $y = e^{\sin x}$ contains an exponential operation and a trigonometric operation, and both are transcendental.
2. Any function containing a transcendental operation is classified as transcendental, so $e^{\sin x}$ is a transcendental function.
3. Regarding elementary functions, the exponential function $e^u$ and the trigonometric function $\sin x$ are both basic elementary functions, and their composition is still an elementary function; the function therefore lies inside the elementary function system while being classified as transcendental.

Answer:

It belongs to transcendental functions because it contains the composition of exponential and trigonometric functions.