%2F(x%2B3).jpeg "(5x-2)/(x+3)")
Exploring the Rational Function (5x-2)/(x+3)
The expression (5x-2)/(x+3) represents a rational function, a function defined as the ratio of two polynomials. Let's dive into its characteristics and explore its behavior.
Key Features
- Domain: The function is defined for all real numbers except for x = -3. This is because the denominator becomes zero at x = -3, leading to an undefined value.
- Vertical Asymptote: At x = -3, the function has a vertical asymptote. This means the function approaches infinity as x approaches -3 from either side.
- Horizontal Asymptote: The degree of the numerator (1) is equal to the degree of the denominator (1). Therefore, the horizontal asymptote is found by dividing the leading coefficients of the numerator and denominator, resulting in a horizontal asymptote at y = 5.
- x-intercept: To find the x-intercept, we set the function equal to zero: (5x-2)/(x+3) = 0. This implies that 5x - 2 = 0, and solving for x gives us x = 2/5. Therefore, the x-intercept is at (2/5, 0).
- y-intercept: To find the y-intercept, we set x = 0: (5(0) - 2)/(0 + 3) = -2/3. Therefore, the y-intercept is at (0, -2/3).
Visual Representation
The graph of this function would show a hyperbola-like shape. It would approach the vertical asymptote at x = -3 and the horizontal asymptote at y = 5. It would intersect the x-axis at (2/5, 0) and the y-axis at (0, -2/3).
Applications
Rational functions like (5x-2)/(x+3) are used in various fields, including:
- Physics: Modelling physical phenomena like the inverse square law of gravity and the relationship between force and distance.
- Engineering: Analyzing circuit behavior and designing control systems.
- Economics: Describing supply and demand curves and modeling economic growth.
Summary
By understanding its key features, graph, and applications, we gain a deeper understanding of the rational function (5x-2)/(x+3) and its relevance in various disciplines.