Help I am so lost....... :(?

Question

State the domain of the following:
a) f(x)= √x+4 sqrt is over both
b) g(x)= (2x+1)/(x-3)
c) h(x)= 3x^2+5x-3
d) l(x)= 2x+3
e) m(x)= (3)/(x^2+7)

If someone could show me step by step how to get to the answer that would be great and thank you to anyone who helps.......

Answer 1

Domain is basically all the values x can be. If you put in something for x and it comes out as a negative square root or dividing by 0 then you know x can't have that value. One way to check this is by graphing the function - you can use a T-chart to graph by hand, or any suitable graphing calculator - and checking whether the graph extends over all x-values or just some of them. You can also, however, solve these problems analytically - that means with algebra, without graphing. Let's do that.

a) Whenever you see a square root in a domain problem, think to yourself, whatever goes under that square root CANNOT come out negative. because you can't take the square root of a negative number. So what would make x + 4 negative? well, if x were -4, that would make x + 4 equal to 0. You know that if x is anything LESS than -4, say -5 or -10, x + 4 comes out negative. So the domain of x is: x can be -4 and anything greater than -4, or x >= -4 (>= means "greater than or equal to").

b) You've got a division sign in this problem. Essentially it's a fraction. Whenever you see division in a domain problem, remember that denominators (the lower part of the fraction) CAN'T be equal to 0. You can't divide by 0. That means x - 3, the denominator in this problem, cannot be 0. How would it be 0? well, if x = 3, x - 3 turns out to be 0. So x CANNOT equal 3. So the domain of x, or all the values x can be, is all real numbers EXCEPT for 3. Usually, when talking about the domain for this kind of thing, you'd just state "x is not equal to 3", and it's understood x can be anything else EXCEPT 3.

c) Well, there's no denominator here, or a square root. Nothing funky. You could test out a couple values for x: -100, maybe, or -5, or 0, or 10.... if you test all of those you'll find you come out with perfectly legitimate answers for y. Well, it turns out this function is what's called a polynomial, and polynomials are generally just valid for all real values of x. so x can be ANYTHING.

d) Same here. It's also a polynomial.

e) denominator once more! remember, x^2 + 7, which is the denominator in this case, cannot be equal to 0. what would make x^2 + 7 equal to 0? well, let's set it equal:

x^2 + 7 = 0
Subtract 7 from both sides:
x^2 = -7
Take the square root of both sides
x = sqrt(-7)
oops! we're taking the square root of a negative number! guess there are NO values of x that will give you 0 in the denominator.... so once more this function is valid for ALL REAL VALUES of x. there's nothing you can put in for x that won't work.

Answer 2

To find the domain of a, think about all possible values of x that will make the problem undefined or imaginary (a negative value under the square root). In this case, as long as x is greater than or equal to 4, the function is defined. So x >= 4 is the domain.

For b, the only time this function will be undefined is when there's a 0 in the denominator. This will only be true when x = 3. So the domain is all real number except 3.

For c and d, the domain is all real numbers because the function is defined for all possible values of x.

For e, you must check (since there is a fraction ) to see if there are any possible values that will give you a 0 in the denominator. In this case it is no because any time you square a number(other than 0) it will be positive and in addition you will be adding 7 to it, so the denominator will always be 7 or higher. Therefore the domain is all real numbers.

Answer 3

Domain is fancy math talk for "what are the possible values of" in this case x so the question becomes what values can x be for :
a. since you have a square root over x and you know that we can only find the square root of positive numbers then x cannot be less than -4. So x is -4 to + infinity. for your square root to be 0 to any other positive number.

b. You know that we can not divide by zero because we don't have an answer for a number divided by 0. So x can be anything from - infinity to + infinity except 3 because it will give a zero denominator which means divide by zero, not allowed.

c. Here x can be anythig you want, no limitations so + infinity to - infinity.

d. same as c.

e. This is a tricky one. the only time this would not work is if x^2 = -7 then you would get a 0 in the denominator which is not allowed, However since x is squared then x^2 will never be negative because if x were negative to begin it becomes + after squaring and so does any positive value of x. So x can be any + or - value. Same as c.

Answer 4

when i state the answers to these problems i am going to be using interval notation. (a,b) a represents the lowest value of the set of numbers i am taking about and b represents the highest values. all of the values in between are included in the set. ( means that the particular value is not included in the set and [ means that the value is.

for example [a, b) refers to all of the real numbers in between a and b on a number line, including a and not including b.



a. use inequality x+4 is greater than or equal to 0 .
this is because you cannot take the square root of a negative number if you are only using the real number system.

DOMAIN: [-4, infinity)

b. DOMAIN: (-infinity, 3) union (3, infinity) or all real numbers not equal to 3

this is because because the denominator of the rational expression (or any fraction) cannot be equal to 0 or the value of the function is undefined

c. DOMAIN: (-infinity, infinity) or all real numbers.

There are no restrictions. The function is defined for all real numbers. this is true for any polynomial.

d. DOMAIN : (-infinity, infinity) the domain is also all real numbers. this is true for any linear equation.

e. the domain is all real numbers because the denominator (x^2 + 7) is never equal to zero for any real numbers. you can check that denominator with the quadratic formula or on your graphing calculator to verify that there are no values which make that parabola equal to 0. if you use the quadratic formula you will no that there are no restrictions on the input of the original function if you find that the solutions to the quadratic equation or all complex numbers. this means that there are no values for which the denominator is equal to zero and the domain of the function is all real numbers.



two important rules to remember for these types of questions are that you can not take the square root of a negative number and the denominator of a fraction can never be equal to zero.

Answer 5

Basicly when you are looking for domain, you are trying to avoid dividing by zero or taking the square root of a negative number. For the first one, in taking the square root of x+4, x+4 must be greater than zero, so x must be greater than negative 4.

In the second, to divide by x-3, x-3 cannot be zero, so x cannot be 3,

(c) and (d) have no restrictions so the domain is all real numbers for both of them. (e) is similar to (b)

Answer 6

wow. thats confusing as crazy.

Answer 7

wow sorry i have no clue how to do that = (