How would I find the domain and range of: k(x)= (1/(4-x^2)+sqrt(x-1)?

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embizze eNotes educator| Certified Educator

Find the domain and range for `k(x)=1/(4-x^2)+sqrt(x-1)` .

(1)The domain is the set of all possible inputs. Generally you assume that the domain is all real numbers (unless the problem involves discrete objects like the number of people) and then find any restrictions. The restrictions include, but are not limited to, not dividing by zero and not taking even roots of negative numbers.

So in this case we check for division by zero; if x=2 then the fraction has a zero in the denominator, so x=2 is not in the domain.

We also check for taking even roots of a negative number: if `x<1` then we are taking a square root of a negative number which we cannot do in the real numbers.So x<1 is not in the domain.

Thus the domain is `x>=1,x!= 2`,or `[1,2)uu(2,oo)` .

(2) The range is the set of all possible outputs. It helps to look at a graph.

Notice that as x approaches 2 from the left, the function grows without bound. This is because for 1.75<x<2, 4-x^2 is smaller than one and approaching zero, so the reciprocal gets arbitrarily large (goes to positive infinity).

As x approaches 2 from the right, for 2<x<2.23 4-x^2 is negative and smaller than 1 in absolute value, so the fraction gets arbitrarily small (goes to negative infinity).

As x gets large, the fraction gets close to zero, but the square root grows without bound.

So the range of the function is all real numbers; the function takes on every value for y at some point.

Thus the domain is `1<=x<2 uu x>2` and the range is all real numbers.