Questions: Determine where the function (m(x)=fracx+4(x-4)(x-6)) is continuous. The function is continuous on (square) (Simplify your answer. Type your answer in interval notation.)

$Determine where the function (m(x)=fracx+4(x-4)(x-6)) is continuous. The function is continuous on (square) (Simplify your answer. Type your answer in interval notation.)$

Transcript text: Determine where the function $\mathrm{m}(\mathrm{x})=\frac{\mathrm{x}+4}{(\mathrm{x}-4)(\mathrm{x}-6)}$ is continuous. The function is continuous on $\square$ (Simplify your answer. Type your answer in interval notation.)

Solution

Solution Steps

To determine where the function $ m(x) = \frac{x+4}{(x-4)(x-6)} $ is continuous, we need to identify the points where the function is not defined. The function is undefined where the denominator is zero, i.e., at $ x = 4 $ and $ x = 6 $. Therefore, the function is continuous everywhere except at these points.

Solution Approach

Identify the points where the denominator is zero.
Exclude these points from the domain of the function.
Express the domain in interval notation.

Step 1: Identify Discontinuities

The function $ m(x) = \frac{x + 4}{(x - 4)(x - 6)} $ is undefined where the denominator is zero. Setting the denominator equal to zero gives us the equations: \[ (x - 4)(x - 6) = 0 \] This results in the discontinuities at $ x = 4 $ and $ x = 6 $.

Step 2: Determine the Domain

The function is continuous everywhere except at the points of discontinuity. Therefore, we exclude $ x = 4 $ and $ x = 6 $ from the domain. The domain can be expressed in interval notation as: \[ (-\infty, 4) \cup (4, 6) \cup (6, \infty) \]