Can someone show me an easy way to solve for Range?

Question

Well, I am having a lot of trouble trying to find one easy way I can use to find range for all problems. I understand how to get domain, but range is just so hard!!

If you can show me the steps to these two problems, I'd GREATLY appreciate it!

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g(t) = 3 - radical (t)

h(t) radical (3 - t) - 7

Answer 1

g(t) = 3 - radical (t)
You can see that t>=0 , else g(t) becomes imaginary.
When t = 0, g(t) = 3, when t = 9 g(t) = 0 and fro then on as t grows larger and larger, g(t) goes more and more negative.So
3>= g(t) > -infinity is the range.

h(t) radical (3 - t) - 7
If t > 3 h(t) becomes imaginary and so must be excluded.
If t= 3, h(t) = sqrt(0) -7 = -7 which is it's most negative value
As t moves left towards - infinity, g(t) gets less and less negativ and finally goes positive and is on its way to + infinity.
Thus the range is g(t) >= -7

Answer 2

When working with functions, we come across two new vocabulary terms: DOMAIN & RANGE. What is a domain? What is a range?

Domain: The domain of a function is the set of all possible input values (usually x), which allows the function formula to work.

NOTE: The notation f(x) is the SAME notation as the letter y. In other words, f(x) = y and y = f(x). The notation f(x) is read "f of x" or "f times x." So, f(x) = x + 4 is the SAME as the expression y = x + 4. Okay, back to domain.

Much of the time a function's domain is all real numbers. However, there can be exceptions. For example, in the function f(x) = 6x/(x-4), x CANNOT equal 4 because to replace x with 4 would yield division by zero and division by zero DOES NOT EXIST. What is the domain of the above function? We can say that the domain of f(x) = 6x/(x-4) is the set of all real numbers x such that x CANNOT equal four. In other words, the domain is ANY NUMBER that does NOT produce division by zero when we plug it in for x. Divisibility by zero is probably the most common way that a function can have a limited domain.

Range: The range is the set of all possible output values (usually y), which result from using the function formula. Keep in mind that domain = INPUT and y = OUTPUT. So we can remember that the point (x,y) = (domain, range). In other words, INPUT = domain, which is x and range = OUTPUT, which is y. Just remember that the domain and range are ALL possible values that can work in the function.

Sample: The function f(x) = x2 will always produce values that are positive or zero because of the squaring process. What is the range of f(x) = x2 ? The range is the set of all real numbers y such that y is greater than or equal to 0. In other words, y (the OUTPUT) will be greater than or equal to 0. What about if we plug in a negative number for x? Even then the OUTPUT will remain positive or greater than zero because of the squaring process. This is what I mean: Say x = -4. f(-4) = (-4)2 = 16, right? The answer will be 16 even if I plug positive 4 for x.

Answer 3

The range is the set of values the answer (g(t)) can have. Since you know the square root of t must be at least zero, 3 minus it must be less than or equal to 3, so that's the range,

Similarly, radical (3-t) must be at least zero, so 0-7 is as low as h(t) can be, so range is all numbers greater than or equal to -7