Is there any value of [tex]$x$[/tex] that would cause this function to be undefined?
If there are restrictions on the domain, explain those restrictions. If there are no restrictions, explain why that is.
Answer (1)
The function F ( x ) = lo g ( x − 5 ) + 1 is undefined when the argument of the logarithm is not positive.
Set up the inequality 0"> x − 5 > 0 to find the valid domain.
Solve the inequality to find 5"> x > 5 .
The function is undefined for x ≤ 5 , so the domain is 5"> x > 5 . 5}"> x > 5
Explanation
Analyzing the function The given function is F ( x ) = lo g ( x − 5 ) + 1 . We need to find any values of x that would make the function undefined and explain any domain restrictions.
Identifying the restriction The logarithm function, lo g ( u ) , is only defined for 0"> u > 0 . Therefore, for F ( x ) to be defined, we must have 0"> x − 5 > 0 .
Solving the inequality To find the restriction on x , we solve the inequality 0"> x − 5 > 0 . Adding 5 to both sides, we get 5"> x > 5 . This means that the function is undefined for any x ≤ 5 .
Determining the domain Therefore, there are restrictions on the domain of x . The function F ( x ) = lo g ( x − 5 ) + 1 is undefined for x ≤ 5 . The domain of the function is all real numbers x such that 5"> x > 5 .
Examples
Logarithmic functions are used in many real-world applications, such as measuring the intensity of earthquakes on the Richter scale or modeling population growth. In finance, logarithms are used to calculate the time it takes for an investment to double at a fixed interest rate. Understanding the domain restrictions of logarithmic functions is crucial for making accurate predictions and avoiding undefined results in these applications. For example, if we are modeling the population growth of a species using a logarithmic function, we need to ensure that the input (e.g., time) results in a positive argument for the logarithm, as a non-positive argument would lead to an undefined or meaningless result.
