State the domain and the range of each function.
22.) f(x)=e^x +3
23.) g(x)=1n(x-1)
in 22 is it e^(x+3) or (e^x) + 3 ?
either case the domain would be the real numbers, since we can use any x we feel like
as far as the range goes:
for e^(x+3) it would be any real > 0
for (e^x) + 3 any real > 3
check your typing for 23, don't know what it means
actually the +3 is separte so it is
f(x)= e^x +3
and the second problem is g(x)= In(x-1)
To determine the domain and range of a function, we need to consider the restrictions and possible outputs of the function.
22.) f(x) = e^x + 3
Domain: The function f(x) involves an exponential term, e^x which is defined for all real numbers. Therefore, the domain of f(x) is (-∞, +∞), meaning it is defined for all real values of x.
Range: The exponential function e^x is always positive since e raised to any real number will be positive. Additionally, we are adding 3 to the result of the exponential function. Therefore, the range of f(x) will be (3, +∞), indicating that any value greater than 3 can be attained.
23.) g(x) = ln(x - 1)
Domain: The function g(x) involves the natural logarithm (ln) of (x - 1). The natural logarithm function is defined only for positive values. Therefore, the expression (x - 1) needs to be greater than 0, which implies that x > 1. Hence, the domain of g(x) is (1, +∞), indicating it is defined for all real numbers greater than 1.
Range: The natural logarithm function returns real outputs for positive inputs. Thus, the range of g(x) will be (-∞, +∞), demonstrating that any real number could be achieved as a result.
In summary:
- f(x) = e^x + 3:
Domain: (-∞, +∞)
Range: (3, +∞)
- g(x) = ln(x - 1):
Domain: (1, +∞)
Range: (-∞, +∞)