How Exactly do you Find the Range of a Function? Am I correct?

Question

Let's take the problem (x^2) - 5
is the Domain: All real #'s except [ sqrt(5), -sqrt(5) ]
(is radical(5) even a real #?) If not, what is the Domain?
and How do you find the Range in general and in this particular problem?

Answer 1

The domain is all reals.

The range is -5 to +infinity.

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The domain is the subset of the reals for which the function is defined. In your function: (x^2) - 5
Any real no. squared and then reduced by 5 will generate a real difference. So your domain ends up being (the improper subset of) all reals. (-infinity, +infinity)

When determining your domain it helps to know that for the BASIC operations (add, subtract, multiply, divide) with reals you'll always get an answer/real EXCEPT when dividing by zero. When taking roots your must avoid taking even roots of negative numbers (it produces a non-real). Odd roots of reals produce reals.

Knowing your domain, then you "test" it in your function and you see from your first term (in the function) you'll never get anything smaller than zero; the first term will always be greater than or equal to zero. So you deduce your range (looking at the second term) -5 to positive infinity, or use the notation: [ -5, +infinity)

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Reals are comprised of all the rational and irrational numbers
(which are disjoint sets). The numbers you gave, sqrt(5),
-sqrt(5) are real irrationals. The non-reals, often called imaginary, are disjoint from the reals but with the reals make up the set of complex numbers which take the form:
a + bi (where a and b are both real).

Answer 2

Ummm... wow, you are really off on this one. The domain of this function is all real numbers, as you could put absolutely anything into this expression and get a number back. I think you are confused about why you couldn't put certain numbers into a function. You could only have this situation if it would cause your function to be undefined (i.e., a negative number under a square root or a denominator equal to zero). Because you have neither here, the domain could be anything.

Now, the range is a bit different. Basically, you just have to know what values would come from this function. In this case, you need to know that this is a parabola opening upward from the point (0,-5). Therefore, the range of the function would be -5 ≤ y < ∞. This could be also be written as [-5 , ∞].

Answer 3

In this case, the domain is the set of all real numbers since there isn't any value you can substitute for x and have the function be undefined. The range of the function is all reals which are ≥ -5 since x²-5 assumes the value of -5 at x = 0 and increases for all x > 0 or < 0.

Doug

Answer 4

the domain for f(x) = xx-5 is all reals. the fact that the function takes on the value 0 does not exclude that point from the domain. The range, however, is all reals >= -5. Since xx is non-negative for real x, the function takes on only (and all) values at least -5