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.
Answer
Examples include $f(x) = 2x + 3$, $f(x) = x^2 + x + 1$, $f(x) = \sin(x)$, and $f(x) = e^x$.
Answer for screen readers
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
- 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.
- 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.
- 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.
- 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.
- 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: $f(x) = e^x$
More Information
These functions are commonly used in mathematics, and they play significant roles in various fields including physics, engineering, and economics. They are characterized by having no restrictions on the input values, making them versatile for analysis and modeling.
Tips
- Confusing functions with limited domains as functions defined for all real numbers. For example, the square root function ( f(x) = \sqrt{x} ) is not defined for negative numbers.
- Forgetting that some functions may appear complex but are still defined for all inputs, such as ( f(x) = \tan(x) ) is only defined where ( x \neq \frac{\pi}{2} + n\pi ) for integers ( n ).