(3x^2+5)/(x^2+4)

4 min read Jun 16, 2024

Exploring the Rational Function: (3x^2 + 5) / (x^2 + 4)

This article delves into the fascinating world of rational functions, specifically analyzing the function (3x^2 + 5) / (x^2 + 4). We'll explore its key features, including:

1. Domain and Asymptotes

Domain: The domain of a rational function is all real numbers except for the values that make the denominator zero. In our case, x^2 + 4 never equals zero (since it's always positive). Therefore, the domain is all real numbers.
Horizontal Asymptote: Since the degree of the numerator and denominator are the same (both are 2), the horizontal asymptote is found by dividing the leading coefficients: 3/1 = 3. This means the function approaches y = 3 as x approaches positive or negative infinity.
Vertical Asymptotes: As the denominator never equals zero, there are no vertical asymptotes.

2. Intercepts

x-intercept: To find the x-intercept, we set y = 0 and solve for x:
- (3x^2 + 5) / (x^2 + 4) = 0
- 3x^2 + 5 = 0
- This equation has no real solutions, meaning the function does not intersect the x-axis.
y-intercept: To find the y-intercept, we set x = 0 and solve for y:
- (3(0)^2 + 5) / (0^2 + 4) = 5/4.
- The y-intercept is at the point (0, 5/4).

3. Symmetry

Even Function: A function is even if f(-x) = f(x). In our case,
- f(-x) = (3(-x)^2 + 5) / ((-x)^2 + 4) = (3x^2 + 5) / (x^2 + 4) = f(x)
- Since f(-x) = f(x), the function is even, meaning it is symmetric about the y-axis.

4. Behavior and Graph

As x approaches positive or negative infinity, the function approaches its horizontal asymptote at y = 3.
The function has a minimum point at (0, 5/4), its y-intercept.
The graph is concave up for all values of x.
The function is always positive, as both the numerator and denominator are always positive.

By understanding these key features, we gain a clear picture of the behavior of the function (3x^2 + 5) / (x^2 + 4) and can accurately sketch its graph.