State the domain and the range of each function.
22.) f(x)=e^x +3
23.) g(x)= In(x-1)
To determine the domain and range of a function, we need to consider the values that x can take and the corresponding values of f(x) or g(x).
For function f(x) = e^x + 3:
- Domain: Since the exponential function e^x is defined for all real numbers, there are no restrictions on the values of x. Therefore, the domain of f(x) is (-∞, ∞), which means that x can take any real number.
- Range: Since the exponential function e^x is always positive, adding 3 to e^x will result in values greater than 3. So, the range of f(x) is (3, ∞), which means that f(x) can take any real number greater than 3.
For function g(x) = ln(x-1):
- Domain: The natural logarithm function ln(x) is defined only for positive values of x. Therefore, in order to evaluate ln(x-1), x-1 must be greater than 0. Solving for x, we find that x > 1. Thus, the domain of g(x) is (1, ∞), which means that x can take any real number greater than 1.
- Range: The natural logarithm function ln(x) returns all real numbers as its output. Since ln(x-1) is a shifted version of ln(x), the range of g(x) will be all real numbers as well. Therefore, the range of g(x) is (-∞, ∞).
To determine the domain, consider any restrictions on x values (such as division by zero or square root of negative numbers). To determine the range, consider the output values that the function can take based on its definition or any transformations applied to it.