# Which function has a domain of all real numbers?

#### Understand the Problem

The question is asking for a mathematical function that is defined for every real number, meaning there are no restrictions on the input values. Examples of such functions include linear functions, polynomial functions, and certain trigonometric functions.

Examples include $f(x) = 2x + 3$, $f(x) = x^2 + x + 1$, $f(x) = \sin(x)$, and $f(x) = e^x$.

Examples of functions defined for every real number include:

• Linear: $f(x) = 2x + 3$
• Polynomial: $f(x) = x^2 + x + 1$
• Trigonometric: $f(x) = \sin(x)$
• Exponential: $f(x) = e^x$

#### Steps to Solve

1. Identify Types of Functions

To find examples of functions defined for every real number, consider linear functions, polynomial functions, trigonometric functions, and exponential functions. For instance, a linear function can be written as ( f(x) = mx + b ), where ( m ) and ( b ) are constants.

1. Linear Function Example

A simple linear function is ( f(x) = 2x + 3 ). This function is defined for all real numbers since you can input any real number ( x ) and get a corresponding output.

1. Polynomial Function Example

A polynomial function of degree 2 is ( f(x) = x^2 + x + 1 ). Like the linear function, this polynomial is defined for all real numbers.

1. Trigonometric Function Example

The sine function ( f(x) = \sin(x) ) is another example of a function defined for every real number. You can input any value for ( x ), and it will give you an output.

1. Exponential Function Example

Consider the exponential function ( f(x) = e^x ). This function is also defined for every real number and produces a positive output for any input.

Examples of functions defined for every real number include:

• Linear: $f(x) = 2x + 3$
• Polynomial: $f(x) = x^2 + x + 1$
• Trigonometric: $f(x) = \sin(x)$
• Exponential: $f(x) = e^x$