4. Find f(x) + g(x) when f(x) = x^(1/2) and g(x) = 6x^(1/2).
===> I got the square root of 7x. Is this right?
16. Find f(x)/g(x) when f(x) = (2x)^(1/2) and g(x) = 2*(sqrt2)*x^(1/3).
===> I got (sqrt2x) / (3*cubertx), but I definitely don't think this is right....if it isn't, can someone show me how to do it please? :)
29. Find the domain of f(f(x)) when f(x) = x^2.
===> f(f(x)) would equal x^4, right? Weeeeeell, how would you graph this to find the domain? I thought it would just be all real numbers greater than zero, but that doesn't seem right -_-
Any help is GREATLY appreciated!!! :D
4. "I got the square root of 7x. Is this right?"
Nearly, but not quite. Careful with your sqrt signs.
f(x)=sqrt(x) g(x)=6sqrt(x)
f(x)+g(x) = sqrt(x) + 6 sqrt(x) = 7 sqrt(x), not sqrt(7x)
16. This is much easier on paper!
Consider your constants. You have sqrt(2) on top, and 2sqrt(2) on bottom. Cancel the sqrt(2)s to leave 2 on the bottom, or 1/2
Consider your variable. You have x^1/2 on top and x^1/3 on bottom. That gives you x^(1/6) - the sizth root of x.
So you end up with 1/2*x^1/6
29. I'm not sure why the domain (not range!) would not be just all real numbers. (The range will always be 0 or positive, of course)
#4 your wrote : "I got the square root of 7x. Is this right?"
If by that you mean 7√x, that would be correct, but the way you expressed it sounds like √(7x).
√(7x) would be wrong.
16. just take (2x)^(1/2) / 2√2x^(1/3)
= x^(1/2 - 1/3) / 2
= x^(1/6) / 2
29.
if f(x) = x^2 , then f(f(x)) = x^4
(you had that)
Now what kind of x's can you put in there ?
Wouldn't any x be allowed ?
So the domain is the set of real numbers.
4. To find f(x) + g(x), you simply add the two functions together. Therefore, f(x) + g(x) would be equal to x^(1/2) + 6x^(1/2). To simplify this expression, you can combine the like terms, which in this case are the terms with the same exponent of 1/2. So, x^(1/2) + 6x^(1/2) can be rewritten as (1 + 6)x^(1/2), which equals 7x^(1/2). Thus, f(x) + g(x) simplifies to 7x^(1/2). Therefore, your answer is correct.
16. To find f(x)/g(x), you divide f(x) by g(x). So, (2x)^(1/2) divided by 2*(sqrt2)*x^(1/3) would be equal to [(2x)^(1/2)] / [2*(sqrt2)*x^(1/3)]. To simplify this expression, you can simplify the numerator and the denominator separately.
Starting with the numerator, (2x)^(1/2) can be simplified as sqrt(2x).
Moving on to the denominator, you need to simplify 2*(sqrt2)*x^(1/3). First, you can simplify sqrt(2) as (sqrt2)^2, which equals 2. Then, x^(1/3) can be simplified as the cube root of x. Therefore, 2*(sqrt2)*x^(1/3) simplifies to 2*(2)*cubertx, which is 4*cubertx.
Now that you have simplified the numerator and the denominator, you can rewrite f(x)/g(x) as sqrt(2x) / (4*cubertx). This is the simplified form of the expression. Thus, your answer is correct.
29. To find the domain of f(f(x)), you need to consider the values of x that are allowed in the function. In this case, f(x) = x^2, which is a simple quadratic function.
When you find f(f(x)), you're essentially applying the function f(x) to f(x) itself. So, you would square the expression f(x), which is x^2. Therefore, f(f(x)) would be equal to (x^2)^2, which simplifies to x^4.
To find the domain of x for f(f(x)), you want to determine the values of x that don't result in any restrictions or undefined behavior. For any quadratic function, there are no restrictions on the domain. This means that any real number can be used as a value for x in the function f(f(x)) = x^4. Therefore, the domain of f(f(x)) is all real numbers. Thus, your answer is correct.