Table of Contents
In this document, we will explore the concepts of domain and range in the context of algebraic functions.
Domain and Range of a Function
- Domain: The set of all possible input values (usually 'x') which the function can accept.
- Range: The set of all possible output values (usually 'f(x)') that the function can produce.
LaTeX notations:
- Domain: \( D(f) \)
- Range: \( R(f) \)
Example 1: Linear Function
Consider a simple linear function \( f(x) = 2x + 3 \).
Domain
- The domain of a linear function is all real numbers, as there are no restrictions on the value of 'x'.
- \( D(f) = {x \in \mathbb{R}} \)
Range
- The range is also all real numbers because a linear function can produce any real value as output.
- \( R(f) = {f(x) \in \mathbb{R}} \)
R Code Example
f <- function(x) { 2*x + 3 }
# Example values
x_values <- c(-10, 0, 10)
y_values <- sapply(x_values, f)
print(y_values)
Example 2: Quadratic Function
Consider a quadratic function \( f(x) = x^2 - 4 \).
Domain
- Similar to a linear function, the domain of a quadratic function is all real numbers.
- \( D(f) = {x \in \mathbb{R}} \)
Range
- The range of this quadratic function is all real numbers greater than or equal to -4, since the lowest value of \( f(x) $\)is -4 (when \( x = 0 \)).
- \( R(f) = {f(x) \in \mathbb{R} | f(x) \geq -4} \)
R Code Example
f <- function(x) { x^2 - 4 }
# Example values
x_values <- c(-3, 0, 3)
y_values <- sapply(x_values, f)
print(y_values)