Question

97977-elementary calc (m)
work: mml #2
question 5, 2.3.37
hw score: 22.67%
points: 0 of 1
determine where the function ( f(x) = \frac{6x}{x^2 + 4} ) is continuous.
the function is continuous on (square).
(simplify your answer. type your answer in interval notation.)

Explanation:

Step1: Analyze the denominator

For a rational function \( F(x)=\frac{6x}{x^{2}+4} \), we need to check when the denominator is zero (since division by zero is undefined). The denominator is \( x^{2}+4 \).
Solve \( x^{2}+4 = 0 \). Rearranging, we get \( x^{2}=-4 \). Since the square of a real number cannot be negative, there are no real solutions for \( x \) that make the denominator zero.

Step2: Determine continuity

A rational function is continuous everywhere except where the denominator is zero. Since the denominator \( x^{2}+4 \) is never zero for real \( x \), the function \( F(x) \) is continuous for all real numbers. In interval notation, this is \( (-\infty, \infty) \).

Answer:

\((-\infty, \infty)\)