Question

Determine where the function \( m(x)=\frac{x+4}{(x-4)(x-7)} \) is continuous. The function is continuous on (Simplify your answer. Type your answer in interval notation.)

Answer 1

The function
`\( m(x)=(x+4)/((x-4)(x-7)) \)` is continuous on the intervals (-∞, 4) ∪ (4, 7) ∪ (7, ∞).

Step-by-step explanation:

A rational function like
`\( m(x)=(x+4)/((x-4)(x-7)) \)` will be continuous wherever its denominator is not equal to zero because division by zero is undefined in mathematics. In this function, the denominators are (x - 4) and (x - 7). To find where the function is continuous, consider the values that make these denominators zero.

The function is discontinuous at x = 4 and x = 7 because these values would make the denominators zero, causing division by zero, which is undefined. Therefore, the function is not continuous at x = 4 and x = 7.

To express this in interval notation:

• - The function is continuous on the intervals to the left of 4 (from negative infinity up to 4), represented as (-∞, 4).
• - It is also continuous between 4 and 7, denoted as (4, 7).
• - Additionally, the function is continuous on the intervals to the right of 7 (from 7 to positive infinity), expressed as (7, ∞).

These intervals are where the function
`\( m(x)=(x+4)/((x-4)(x-7)) \)` remains continuous, excluding the points where the denominators become zero. Hence, the function is continuous on the intervals (-∞, 4) ∪ (4, 7) ∪ (7, ∞).