What is the parent function of a rational function?
The Correct Answer and Explanation is :
The parent function of a rational function is defined as ( f(x) = \frac{1}{x} ). This function serves as the foundational model for all rational functions, which can be expressed as the quotient of two polynomial functions, ( f(x) = \frac{p(x)}{q(x)} ), where ( p(x) ) and ( q(x) ) are polynomials.
Characteristics of the Parent Function ( f(x) = \frac{1}{x} )
- Domain and Range: The domain of ( f(x) = \frac{1}{x} ) excludes zero, as division by zero is undefined. Thus, the domain is ( x \in (-\infty, 0) \cup (0, \infty) ). The range is also ( y \in (-\infty, 0) \cup (0, \infty) ), meaning the function can take any positive or negative value but never zero.
- Asymptotes: This function has two asymptotes:
- Vertical Asymptote: At ( x = 0 ), where the function approaches infinity as ( x ) approaches zero from the right and negative infinity as ( x ) approaches zero from the left.
- Horizontal Asymptote: As ( x ) approaches positive or negative infinity, ( f(x) ) approaches zero.
- Graph: The graph of ( f(x) = \frac{1}{x} ) consists of two distinct branches located in the first and third quadrants of the Cartesian plane. The curve is symmetric about the origin, demonstrating odd symmetry (i.e., ( f(-x) = -f(x) )).
- Transformations: Any rational function can be derived from this parent function by applying transformations such as vertical and horizontal shifts, reflections, stretches, and compressions. For example, the function ( f(x) = \frac{1}{x – 2} + 3 ) represents a horizontal shift to the right by 2 units and a vertical shift upward by 3 units.
Overall, understanding the parent function ( f(x) = \frac{1}{x} ) provides a solid foundation for studying more complex rational functions, enabling a better grasp of their behavior and characteristics.