Ohio Assessments for Educators (OAE) Mathematics Practice Exam

What is a defining characteristic of a transcendental function?

1. It can be represented by a polynomial equation

2. It must include logs, trigonometric functions, or variables as exponents

3. It can be expressed in a finite number of terms

4. It always has a real number as its output

The correct answer is: It must include logs, trigonometric functions, or variables as exponents

A defining characteristic of a transcendental function is that it must include logs, trigonometric functions, or variables as exponents. Transcendental functions, such as exponential functions, logarithmic functions, and trigonometric functions, cannot be expressed as finite polynomials. This distinction is crucial because while algebraic functions can be defined by polynomial equations (which would mean they can be represented in a finite number of terms), transcendental functions go beyond these algebraic expressions. For instance, the function \( e^x \) is an exponential function, and it does not have a representation that can be encapsulated by a polynomial equation. Similarly, a function like \( \sin(x) \) is a trigonometric function which also doesn't fit into polynomial criteria. This fundamental difference marks transcendental functions and highlights their broader application and properties in mathematics compared to polynomial functions. Transcendental functions can take on a wide variety of forms and properties, making them highly versatile and applicable in various contexts, such as calculus, physics, and engineering. Understanding the nature of these functions helps in correctly identifying and applying them in mathematical reasoning.