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The question in the image asks:

**"What is an algebraic function? Transcendental function?"**

### Definitions:

1. **Algebraic Function**:
   - An algebraic function is a function that can be defined as the root of a polynomial equation whose coefficients are themselves polynomials. In simpler terms, it includes operations like addition, subtraction, multiplication, division, and taking roots (e.g., square roots).
   - Examples include: \( f(x) = x^2 + 3x + 2 \), \( g(x) = \sqrt{x} \), and \( h(x) = \frac{x^3 - 2x}{x - 1} \).

2. **Transcendental Function**:
   - A transcendental function is a function that is not algebraic. These functions cannot be expressed as roots of polynomial equations with a finite number of algebraic operations. Transcendental functions include functions like exponential, logarithmic, trigonometric, and inverse trigonometric functions.
   - Examples include: \( f(x) = e^x \), \( g(x) = \sin(x) \), and \( h(x) = \ln(x) \).

Would you like more details or have any questions? Here are some related questions for further exploration:

1. How do you determine if a given function is algebraic or transcendental?
2. What are some real-world applications of algebraic functions?
3. How do transcendental functions appear in calculus and analysis?
4. What is the significance of transcendental numbers, and how do they relate to transcendental functions?
5. How do algebraic and transcendental functions differ in their graph behavior?

**Tip:** Understanding these functions is crucial in higher mathematics, as algebraic functions often simplify problems, while transcendental functions frequently model real-world phenomena such as waves, growth, and decay.

What is an algebraic function? Transcendental function?

Explore the differences between algebraic and transcendental functions. Learn how algebraic functions are defined as roots of polynomials, while transcendental functions include exponential, logarithmic, and trigonometric functions. Suitable for advanced high school and college students.

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