f(t)=log(t-5)
t-5>0
t>5
g(x)=5e^x
all real numbers
g(x)=In(t+4)
t+4>0
t>4
g(t)=5^t
all real numbers
all ok
They are all right
To determine the domain of a given function, we need to understand the restrictions or conditions that would prevent the function from being defined or evaluated.
1. f(t) = log(t-5):
The domain of this function is determined by the logarithm's argument, which must be greater than zero. In other words, for the expression t-5 > 0, we solve for t to find the values that satisfy the inequality. By adding 5 to both sides of the inequality, we get t > 5. Therefore, the domain of f(t) is all real numbers greater than 5.
2. g(x) = 5e^x:
Exponential functions are defined for all real numbers, so there are no restrictions or conditions on the domain of this function. In other words, g(x) is defined for all real numbers.
3. g(x) = ln(t+4):
The natural logarithm function requires a positive argument, which means that t+4 > 0. By subtracting 4 from both sides of the inequality, we obtain t > -4. Hence, the domain for g(x) is all real numbers greater than -4.
4. g(t) = 5^t:
Exponential functions with positive bases, such as 5^t, are defined for all real numbers since any real number raised to any power will yield a real number. Therefore, the domain of g(t) is all real numbers.
Remember, when determining the domain of a function, it is essential to consider any restrictions implied by the mathematical operations or functions used in the expression.