**Recognizing a Real-Valued Function and a Real Function**

A function is a mathematical concept that describes the relationship between two sets of numbers. In particular, a real-valued function maps real numbers from one set (called the domain) to another set of real numbers (called the range). On the other hand, a real function is a more general term that encompasses functions defined on real numbers, but not necessarily mapping to real numbers. In this guide, we will explore how to recognize a real-valued function and a real function in detail.

**Real-Valued Function:**

A real-valued function is a function whose output values are real numbers. In other words, the range of the function consists solely of real numbers. Here are some key characteristics to identify a real-valued function:

1. **Domain and Range:** A real-valued function has a domain consisting of real numbers. The input values of the function can be any real number. Meanwhile, the range of the function is also composed of real numbers.

2. **Graph:** The graph of a real-valued function lies entirely in the Cartesian plane, where both the x-axis and y-axis represent real numbers.

3. **Examples:** Some common examples of real-valued functions include polynomial functions, exponential functions, logarithmic functions, trigonometric functions, and rational functions.

**Real Function:**

A real function is a broader term that encompasses functions defined on real numbers but does not necessarily map to real numbers. Here are some key characteristics to recognize a real function:

1. **Domain:** A real function has a domain consisting of real numbers. The input values of the function can be any real number.

2. **Range:** Unlike a real-valued function, a real function does not have a specific range requirement. The output values of a real function can be real numbers, complex numbers, or even vectors, matrices, or functions themselves.

3. **Graph:** The graph of a real function can be represented in various ways. It may lie in different coordinate systems, such as the Cartesian plane, polar coordinate system, or even higher-dimensional spaces.

4. **Examples:** Some examples of real functions include complex-valued functions, vector-valued functions, matrix-valued functions, and functional-valued functions.

In conclusion, a real-valued function is a specific type of real function that maps real numbers to real numbers. Recognizing a real-valued function involves ensuring that both the domain and range consist solely of real numbers, while a real function is a more general term that encompasses functions defined on real numbers without any specific range restriction.