SOLUTION: Construct a rational function g(x) such that its natural domain is all of R and its range is given
by the interval [−1, 0)
Algebra.Com
Question 958989: Construct a rational function g(x) such that its natural domain is all of R and its range is given
by the interval [−1, 0)
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
- For the function to be a true rational function, it must take the form:
where n(x) and d(x) are polynomial functions of x and d(x) is not a constant function. - For the function to have a natural domain of all real numbers, d(x) must be a function with no (real) zeros.
- For the function to have a range which does not include zero, n(x) must be a function with no (real) zeros.
- For a function to have a range which includes -1, there must be at least one x for which n(x) = -d(x).
- For the remainder of the range, both of the following must be true for all x's excepts the one(s) which make g(x) = -1:
- |n(x)| < |d(x)|
- the signs of n(x) and d(x) are opposites.
Putting all this together I came up with:
This is not the only possible solution. For example, the 1's in the function above can be replaced with any positive number, And I imagine there may be many additional functions which are significantly different from this one and still meet the requirements of the problem.
RELATED QUESTIONS
Graph the function g(x)=log base 4(x+3) and given its domain and range using interval... (answered by Fombitz)
Graph the function g(x) = 4^(x-1) give its domain and range using interval... (answered by stanbon)
Graph the function {{{ g(x)=4^x-3 }}}
Give its domain and range using interval... (answered by richard1234)
1.Solve
x^2+3x−4/2x^2+−3≤ 0, indicating your answer by interval... (answered by solver91311)
Construct a function (the actual formula for the function, not its graph) whose domain is (answered by Fombitz)
Use a graphing utility to graph the function and estimate its domain and range. Then find (answered by Fombitz)
graph the function g(x)=log base 4 (x+3) and give its domain and range using interval... (answered by Fombitz)
graph the function g(x) = log bottom right 4 (x+3) and give its domain and range using... (answered by Fombitz)
graph the function g(x) = log base10 (x+3) and give its domain and range using interval... (answered by Fombitz)