Domain and Range - algebra?

Question

I really lack a poor understanding of this topic. In a way I understand it,
but in another way I don't. I am really stumped by a few questions. I've
been over my textbook time and time again, and its not clicking.

1. Does y = |x|-5 and y= |2x|-5 have the same domain and/or range?

2. Does y = -x^2+5 and y = -1/2x^2+5 have the same domain and/or range?

3. What is the domain and range of y = sqrt2x+1?

for #3,
domain options -
all real numbers
all real numbers > or = to negative 1/2
all positive real numbers
all real numbers except negative 1/2

range options -
all real numbers
all real numbers < or = to 4
all real numbers > or = to 0
all real numbers except negative 1/2

Answer 1

1) Yes, the domain and range are the same.
2) Yes, the domain and range are the same.
3) Depends on what is underneath the root sign
y = sqrt(2x+1)
For domain, set whatever is underneath the root sign greater than or equal to zero.
y = sqrt(2x+1)
2x + 1 >= 0 means x >= -1/2 (all numbers greater than or equal to -1/2)
For range, the sqrt is never negative.
For y = sqrt(2x+1)
The range is all real number greater than or equal to zero.

Answer 2

OK, just done a little background research, because of vocabulary issues... didn't know the significance of these two words in algebra...
Here we go: what you have there are not equations, but functions. When you define a function, you must specify three things:
1) the DOMAIN --> from where the function gets its input;
2) the RANGE --> to where the function places its output;
3) the RELATION --> (this could be seen as the function itself) it describes the relationship between input and output, or how you get from the first to the second... There are several ways to do this: 3a) ANALITICAL --> you have an expression (like in your examples); 3b) OPERATIONAL --> you explain the steps (i.e. operations) your function has to perform on the input to obtain the output (this has much to do with the notions of algorithm, programming and computers); 3c) SYNTHETICAL (I would prefer BULK ;) --> means providing a correspondence table; for each possible value in the domain (input) you give the corresponding value in the range (output).
Enough with this... OK, some examples... Let's take the basic functions that appear in your problems:
1) the Modulus: y = | x | : R --> R+
DOMAIN: All real numbers.
RANGE: All real positive numbers, zero included.
EXPRESSION: returns x, if x positive of zero, or -x otherwise.
2) the Power: y = x ^ a : R --> R, where a is a Natural number, conventionally x ^ 0 = 1, for any non-zero value of x (in your example this function particularizes for a = 2).
DOMAIN: All real numbers.
RANGE: All real numbers (in general); All real positive numbers, zero included (in your case).
EXPRESSION: returns the second power of x.
3) the Square Root: y = sqrt (x) : R+ --> R+ (I'll cut short the discussion about the general Root function).
DOMAIN: All real positive numbers, zero included.
RANGE: All real positive numbers, zero included.
EXPRESSION: the square root...
In order to establish the boundaries for the domain and range of a composed function, it's useful to compute the function's value at certain key points. Almost always such a point is x = 0. Also helps to know some properties of the basic function (symmetry, monotony etc, singular points...)

But, to give the answers to your questions and a short explanation:
1) They have both the same domain --> All real numbers, AND the same range --> [ -5 ; + Inf. ). These are conditioned by the modulus, a symmetric, positive and monotone-increasing function. The modulus has a unique minimum point at x = 0, where y(0) = | 0 | = 0. Computing the values of both your functions at x = 0, you get: y(0) = | 0 | - 5 = -5 and y(0) = | 2 * 0 | - 5 = -5 respectively. So, -5 is their minimum value, because from this the functions only grow.
2) Yes. Again, they both have the same domain --> All real numbers, AND the same range --> [ 5 ; - Inf. ). The second power function is also a symmetric, positive and monotone-increasing function, with a unique minimum point at x = 0, where y(0) = 0 ^ 2 = 0. But because the coefficient of x is negative, it becomes a symmetric, negative and monotone-decreasing function, and the point of minimum becomes a maximum. Computing the values of both your functions at x = 0, you get: y(0) = - 0^2 + 5 = 5 and y(0)= -1/2 * 0^2 + 5 = 5 respectively. So, 5 is their maximum value, because from this the functions only reduce themselves.
3) The domain is given by the Square Root, which cannot accept negative numbers. So, the correct option is "C" -- All positive real numbers. For the range, take into account that the Square Root function is a positive, monotone-increasing function, with a unique minimum point at x = 0, where it's value is also 0 (zero). From this you go "up" a unit... The completely bounded range would have been [ 1 ; + Inf. ), but this is not among your options... From these available options, I'll choose again "C" -- All greater than or equal to zero (positive) real numbers. Option "A" is too permissive, option "B" is unjustifiable restrictive (the function Square Root has values greater than 4 -- sqrt(25) = 5) and option "D" is less restrictive than "C" for no particular reason -- The Square Root has a problem with ALL negative arguments, not the particular -1/2
Hope you'll find this helpful (if you take the time to read it :)). It would have been more explicitly, it I could've used sketches... Take care !

Answer 3

Domain is just what numbers you are allowed to plug into the equation where as range is what numbers you can get the equation to pump out.

for #3 your domain is all real numbers > or = to -1/2. The reason is you can't take the square root of a negative number.
The range would be any number y can equal provided the constraints we just gave to x. So the range would be all real numbers > or = to 0.

Answer 4

Domain is X and Range is Y
What can X be, this is the domain. What can y be? This is the range.
if y= x^2 x can be anything, the domain has no boundries.
But Y can never be negative, since a negative squared is positive. so the range is >or = 0

Answer 5

It would be helpful , if u have a graphing calulater
Just put in the equation, see how the graph looks like and figure out domain ang range.
Domain-left to right
Range - bottom to top