State the domain and the range of each function.
22.) f(x)=e^x +3 the +3 is separated
23.) g(x)= 1n(x-1)
the first equation I had already answered for you.
domain: any real number
range: any real number > 3
second one, g(x) = ln(x-1)
domain: any real x , where x > 1
range : any real number
You should graph these to see why the answers are the way they are.
Thanks
To determine the domain and range of a function, we need to analyze the behavior and restrictions of the function.
22. f(x) = e^x + 3
- Domain: The domain of this function is the set of all real numbers because there are no restrictions or exclusions mentioned in the function. Therefore, every real number can be plugged into the function.
- Range: The range of this function is the set of all real numbers greater than 3. Since exponential functions like e^x grow indefinitely as x increases, and we add 3 to the result, the range extends infinitely in the positive direction.
23. g(x) = ln(x - 1)
- Domain: In the natural logarithmic function ln(x), the input value x must be greater than 0. Therefore, in the given function g(x) = ln(x - 1), the expression (x - 1) must be greater than 0. So, we set (x - 1) > 0 and solve for x: x > 1. Consequently, the domain of this function is all real numbers greater than 1.
- Range: The range of the natural logarithmic function ln(x) consists of all real numbers. However, to find the range of g(x) = ln(x - 1), we need to consider that the input value (x - 1) must be greater than 0. As a result, the range of this function is all real numbers (or (-∞, ∞)).
In summary:
Function 22:
Domain: All real numbers
Range: All real numbers greater than 3
Function 23:
Domain: All real numbers greater than 1
Range: All real numbers