%2F(x%5E2%2B4).jpeg "(3x^2+5)/(x^2+4)")
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.