|
Question 157489: Hi, I need help getting the domain and range(or just the domain) of this piecewise-defined function f(x)
.
f(x) = { x - 3,  }
........ {  ,  }
........ { 20, x > 12 }
.
Please help me with finding the domain and range of the function, Please use interval notation
.
Thanks ahead of time,Levi
Answer by gonzo(654)   (Show Source):
You can put this solution on YOUR website! domain is the set of all x value that y can be calculated from.
range is the set of all y values that can be calculated.
equation 1.............................................................
f(x) = x-3; x<=4.
they tell you what the domain is i.e. x<=4.
since the biggest number in the domain would be x=4, at that point, y = x-3 = 1, so the range is y <=1.
equation 2.....................................................
f(x) = x^2; 4
the domain is, again, given as 4
to find the range, calculate for the smallest value of x and the greatest value of x.
when x = 4, y would equal 16, so if x must be greater than 4, y must be greater than 16.
when x = 12, y = 12^2 = 144 so if x must be <= 12, than y must be <= 144.
based on the equation, the middle values of x will yield middle values of y, i.e. none of the middle values of y will be less than 12 or greater than 144.
the range is therefore y must be > 16 and <= 144 which can be rewritten as
16 < y <= 144.
equation 3..............................................................
f(x) = 20; x>12.
here the domain is again given as x>12. x has to be > 12.
the range will always be 20 since the function does not allow any variation, i.e. the y value is set to 20 regardless of the value of x.
so the range is y = 20.
............................................................................
if we look at the whole function, then we see the following:
x can be <=4.
x can be > 4 and <= 12
x can be > 12.
what this says is that x can be any value.
so the domain of the whole function taken together is all x (no restrictions) with the implication being that all x means all real number (no imaginary numbers such as (-1)^(1/2). that shouldn't happen because there is no square root function in the domain of x.
.........................................................................
looking at the whole function for the range, then we see the following:
y can be <= 1.
y can be > 16 and <= 144
y can be 20
since 20 is within the range of >16 and <= 144, it doesn't need to be repeated, so the full range is y <= 1 and 16 < y <= 144.
|
|
|
| |
