determine the intervals where f(x) = ln(6x-24) is continuous
ln(u) is defined only where u > 0
So, ln(6x-24) is defined only where 6x-24 > 0. That is, where x > 4.
f(x) is continuous in that domain.
Well, determining intervals where a function is continuous can be quite confusing, but don't worry, I'm here to put a smile on your face while we figure it out!
The natural logarithm function, ln(x), is continuous for all positive values of x. So let's set the argument of ln(6x - 24) greater than zero:
6x - 24 > 0
Solving this inequality, we get:
6x > 24
x > 4
Therefore, the function f(x) = ln(6x - 24) is continuous for all x values greater than 4. Let's have a good laugh while we celebrate math and continuity together! 🎉
To determine the intervals where the function f(x) = ln(6x-24) is continuous, we need to consider the domain of the function and whether there are any discontinuities.
First, let's determine the domain of the function which is the set of all allowable values for x in the expression ln(6x-24).
The natural logarithm function ln(x) is defined only for positive real numbers, so the expression inside the ln function, 6x-24, must be greater than zero.
6x - 24 > 0
Adding 24 to both sides:
6x > 24
Dividing both sides by 6:
x > 4
So, the domain of f(x) = ln(6x-24) is x > 4.
Now, let's look for any possible discontinuities.
Since ln(6x-24) contains only algebraic operations and logarithmic functions, and none of them have any restrictions or boundary conditions, the function has continuous behavior except for the x = 4.
Therefore, the intervals where f(x) = ln(6x-24) is continuous are (4, ∞).
To determine the intervals where f(x) = ln(6x-24) is continuous, we need to consider two main factors: the domain of the function and any potential points of discontinuity.
1. Determine the Domain:
The natural logarithm function, ln(x), is defined only for positive values of x. So, for f(x) = ln(6x-24) to be defined, the expression inside the logarithm must be positive.
6x - 24 > 0
Solving the inequality, we have:
6x > 24
x > 4
Therefore, the domain of f(x) is x > 4.
2. Potential Points of Discontinuity:
Points of discontinuity occur when the function is not defined or when it jumps or breaks. In the case of the natural logarithm function, ln(x), it is defined for all positive values.
However, as we have determined that 6x - 24 > 0, x > 4. This means that the function f(x) = ln(6x-24) is defined only when x > 4. Therefore, there are no points of discontinuity within the given domain.
In conclusion, the function f(x) = ln(6x-24) is continuous for all x in the interval (4, +∞).