Question

Determine where the function \( F(x)=\frac{7 x}{x^{2}+25} \) is continuous. The function is continuous on (Simplify your answer. Type your answer in interval notation.)

Answer 1

The function
`\( F(x) = (7x)/(x^2 + 25) \)` is continuous for all real numbers
`\( x \)` since both the numerator and the denominator are polynomials and the denominator is never equal to zero. Therefore, the function is continuous on the entire real number line, expressed as
`\( (-\infty, \infty) \)`.

Step-by-step explanation:

To determine the continuity of the function
`\( F(x) = (7x)/(x^2 + 25) \)` , we need to check for three conditions: the function must be defined at
`\( x \),` the limit of
`\( F(x) \)` as
`\( x \)` approaches
`\( a \)` from both sides must exist, and the limit must equal the value of
`\( F(x) \) at \( x = a \)` .

In this case, the fu

nction is defined for all real numbers since the denominator
`\( x^2 + 25 \)` is never equal to zero. Additionally, the numerator
`\( 7x \)` and the denominator are both polynomials, ensuring that the limit exists as
`\( x \)` approaches any real number. The function is continuous everywhere, meeting all conditions.

Expressing the interval notation in terms of continuity, we use the symbol
`\( (-\infty, \infty) \)` to denote the entire real number line. This indicates that the function
`\( F(x) \)` is continuous for all values of
`\( x \)` without any breaks or discontinuities. Understanding the conditions for continuity and applying them to the given function provide insights into the behavior of the function across its entire domain.