Functions (6 problems)?

Question

For each of the following, give an example or show that no example can exist.
(a) 2 different functions, f and g, that have the same domain.
(b) 2 different functions, f and g, that have the same range.
(c) 2 different functions, f and g, that have the same domain and same range.
(d) 2 different functions, f and g, such that the range of f is a proper subset of the domain of g.
(e) A function w/a bounded interval for its domain and a bounded interval for its range.
(f) A function w/an unbounded interval for its domain and a bounded interval for its range.

You don't have to answer all of them. At least one would be really helpful.

Answer 1

(a) f(x) = √x, g(x) = √(2x)



(b) f(x) = |x|, g(x) = 2|x|



(c) f(x) = √x, g(x) = 2√x

Answer 2

a) f(x)=x, g(x)= x+1 , both have same domain R

b) the above functions also have the same range

c)see above

d) f(x)=1 , a const function, g(x)= anything.... 1 belongs to domain of g

e)f(x)=x, where x belongs to 0 to 1

f) f(x)=1/(x!), the domain is N, but range is from [0,1]

Answer 3

a. f(x) = x + 2 and g(x) = x - 4

b. f(x) = ln(x) and g(x) = x

c. f(x) = x and g(x) = x^3

d. f(x) = sqrt(x) and g(x) = |x|

e. Can't happen in my knowledge. The only thing that fits this would be a circle, which has an equation, but it is not a function.

f. f(x) = 3 ---> I think I'm just trying to be sneaky and cheap with this one. Let me know what you think.