Give the domain of each of the functions listed below :
a) f(x)= ln (tanh^-1 (x))
b) g(x) = cos ^-1 (e^x)
I just don't know how to set them up to solve for x ...I'm doing a Calc course online on my own, and would really appreciate the help :)
Thanks
a) There can be no log of a negative number or zero, so
tanh^-1(x) > 0
That requires x > 0, since
tanh(x) > 0 for x > 0
b) e^x must be between -1 and 1, since the cosine function is between those extremes.
e^x cannot be <0 for any x
if e^x => 0, x => -infinity
If e^x = 1
x = ln(1) = 0
-infinity < x < 0
To find the domain of a function, we need to consider the restrictions on the values that the input variable, in this case, 'x', can take. Let's break down each function to determine their domains.
a) f(x) = ln(tanh^-1(x))
To find the domain of f(x), we'll start by addressing the innermost function, tanh^-1(x). The hyperbolic tangent inverse function, tanh^-1(x), is only defined for input values between -1 and 1. Therefore, the domain of tanh^-1(x) is (-1, 1).
Next, we take the natural logarithm, ln(), of the output of tanh^-1(x). The natural logarithm, ln(), is defined only for positive real numbers. So, for ln(tanh^-1(x)), we need the output of tanh^-1(x) to be greater than 0.
Therefore, to determine the domain of f(x), we need to find the values of x that satisfy both conditions:
-1 < x < 1 (to satisfy the domain of tanh^-1(x))
tanh^-1(x) > 0 (to satisfy the domain of ln())
Combining these conditions, we can conclude that the domain of f(x) is the set of all real numbers x such that:
-1 < x < 1 and tanh^-1(x) > 0.
b) g(x) = cos^-1(e^x)
For g(x), we have the inverse cosine function, cos^-1(). The inverse cosine function is only defined for input values in the range [-1, 1].
Inside the inverse cosine function, we have e^x. The exponential function, e^x, is always positive for any real value of x. Therefore, its range is (0, ∞).
Since cos^-1() needs input values between -1 and 1, in order to define the domain of g(x), we need to find the values of x for which e^x is between -1 and 1.
However, since e^x is always positive, it can never be between -1 and 1. In this case, there is no real value of x that satisfies this condition, and therefore the domain of g(x) is the empty set (∅). In other words, g(x) is not defined for any x.
To summarize:
a) The domain of f(x) is the set of all real numbers x such that -1 < x < 1 and tanh^-1(x) > 0.
b) The domain of g(x) is the empty set (∅).