Which of the following fuunction(s) is/are transcendal?

Options

A. `f(x)=5sin(sqrt(x))`
B. `f(x)=(2sin3x)/(x^(2)+2x-1)`
C. `f(x)=sqrt(x^(2)+2x+1)`
D. `f(x)=(x^(2)+3)*2^(x)`

Answer

To determine which of the given functions are transcendental, we need to understand the definition of transcendental functions. A transcendental function is a function that is not algebraic, meaning it cannot be expressed as a root of a polynomial equation with integer coefficients.

Let's analyze the options step by step:

### Step 1: Identify the functions
Assume the functions are:
- Option A: \( f(x) = e^x \)
- Option B: \( g(x) = \sin(3x) \)
- Option C: \( h(x) = \sqrt{x^2 + 2x + 1} \)
- Option D: \( k(x) = x^2 + 3 \cdot 2^x \)

### Step 2: Analyze Option A
- **Function**: \( f(x) = e^x \)
- **Analysis**: The exponential function \( e^x \) is a well-known transcendental function because it cannot be expressed as a solution to any polynomial equation with integer coefficients.
- **Conclusion**: Option A is transcendental.

### Step 3: Analyze Option B
- **Function**: \( g(x) = \sin(3x) \)
- **Analysis**: The sine function is also transcendental as it cannot be expressed as a polynomial equation. Therefore, this function is transcendental.
- **Conclusion**: Option B is transcendental.

### Step 4: Analyze Option C
- **Function**: \( h(x) = \sqrt{x^2 + 2x + 1} \)
- **Analysis**: We can simplify this function:
\[ h(x) = \sqrt{(x + 1)^2} = |x + 1| \]
This is an algebraic function because it can be expressed as a polynomial.
- **Conclusion**: Option C is not transcendental.

### Step 5: Analyze Option D
- **Function**: \( k(x) = x^2 + 3 \cdot 2^x \)
- **Analysis**: The term \( 2^x \) is transcendental, and the sum of an algebraic function \( x^2 \) and a transcendental function \( 3 \cdot 2^x \) is also transcendental.
- **Conclusion**: Option D is transcendental.

### Final Conclusion
The transcendental functions among the options are:
- Option A: \( e^x \) (transcendental)
- Option B: \( \sin(3x) \) (transcendental)
- Option C: \( \sqrt{x^2 + 2x + 1} \) (not transcendental)
- Option D: \( x^2 + 3 \cdot 2^x \) (transcendental)

Thus, options A, B, and D are transcendental functions.