Suppose f is a function with range (-∞,∞) and g is a function with domain (0, ∞). Is it possible that g and f are inverse functions? Explain.
I'd say no. The range of a function is the domain of its inverse.
g(f) = x
the domain of g is the range of f.
To determine if g and f can be inverse functions, we need to consider the conditions for two functions to be inverses of each other.
For two functions to be inverses, the composition of the functions should yield the identity function. In other words, if we apply one function and then the other, we should get back the original input value.
Let's assume that g and f are inverse functions. Then, for any input x in the domain of g, applying f(g(x)) should give us x.
However, g(x) gives us an output in the range of f, which is (-∞, ∞). Since f has a range of (-∞, ∞), it means that for any output value of g, there exists a corresponding value in the range of f.
But, the range of f is not restricted to positive numbers; it includes negative numbers and all real numbers as well. Therefore, if we have a negative number as the output of g, there won't be a corresponding input value for f to produce that negative number, violating the condition for inverse functions.
Hence, it is not possible for g and f to be inverse functions because g has a domain of (0, ∞) and f has a range of (-∞, ∞).