%2F(x-5).jpeg "(x+3)/(x-5)")
Exploring the Rational Function: (x+3)/(x-5)
The function (x+3)/(x-5) is a rational function because it's defined as a ratio of two polynomials. Understanding its properties and behavior requires analyzing different aspects.
Domain and Range
The domain of a function is the set of all possible input values (x-values) for which the function is defined. The function (x+3)/(x-5) is undefined when the denominator is zero, which occurs at x=5. Therefore, the domain is all real numbers except for 5.
Domain: x ∈ ℝ, x ≠ 5
The range of a function is the set of all possible output values (y-values). This function can take on any real value except for y=1. This is because the numerator and denominator can never be equal, meaning the function will never simplify to 1.
Range: y ∈ ℝ, y ≠ 1
Asymptotes
Vertical Asymptote: The line x=5 is a vertical asymptote because the function approaches positive or negative infinity as x approaches 5.
Horizontal Asymptote: The line y=1 is a horizontal asymptote. This is because the degree of the numerator and denominator are the same (both degree 1), and the leading coefficient of the numerator (1) divided by the leading coefficient of the denominator (1) equals 1.
Intercepts
x-intercept: To find the x-intercept, set the function equal to zero and solve for x:
(x+3)/(x-5) = 0 x + 3 = 0 x = -3
Therefore, the x-intercept is (-3, 0).
y-intercept: To find the y-intercept, set x equal to zero and solve for y:
(0+3)/(0-5) = -3/5
Therefore, the y-intercept is (0, -3/5).
Graphing the Function
By plotting the intercepts, asymptotes, and analyzing the behavior of the function for different x-values, you can create a graph of the function. It will show a hyperbola-like shape with the vertical asymptote at x=5 and the horizontal asymptote at y=1.
Conclusion
Understanding the key features of rational functions like the domain, range, asymptotes, and intercepts helps visualize and interpret their behavior. These characteristics are important in various applications across mathematics, physics, and engineering.