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## Algebra I (2018 edition)

### Course: Algebra I (2018 edition)>Unit 6

Lesson 5: Introduction to the domain and range of a function

# What is the range of a function?

The range of a function is the set of all possible outputs the function can produce. Some functions (like linear functions) can have a range of all real numbers, but lots of functions have a more limited set of possible outputs.

## Want to join the conversation?

• If g(x)=x^2/x
and
x^2/x=x
then can we say that:
g(x)=x
And if we can, how than for x=0 this expression is undefined?

In Serbia, we learned that expressions needs to be simplified if it is possible. So x^2/x is same as x, and regular way to describe it is to write it like that, just x.

I`m interested in this, because if this is the case with functions, any expression (X) could be writen like that - X^2/X which then means that no expression (X) is defined for X=0, regardless of what expression is.

For example, F(x)=345x is defined for input value of 0, and has output of 0. But 345x=(345x)^2/345x, it is undefined for x=0.

Im sorry if this is stupid question, and if i missed some basic rule with functions, but this is not clear to me. Can we transform expressions in functions like we do in other mathematical expressions or can we not?
• When you start with a reciprocal function, you will have at least one vertical asymptote in which the function does not have a value. So by starting with g(x) = x^2/x, you have a vertical asymptote at x=0, so from the start of your problem, x cannot equal to zero. So when you reduce it to g(x)=x, you have to say that x=0 is an extraneous solution (see https://en.wikipedia.org/wiki/Extraneous_and_missing_solutions for definition of extraneous solution). Hope this answers your question.
• OK so I'm totally lost by this. Is this at all like the domain on the function? They seem totally different but also like the exact same thing.
• The domain of a function is the set of all acceptable input values (X-values).
The range of a function is the set of all output values (Y-values).
Hope this helps.
• what is the difference between f(x) and y values?
• f(x) is Y.
The equation `f(x) = 2x - 5` is the same a `y = 2x - 5` provided the equation with Y is a function. Not all equation are functions. So, you can't always swap out Y for f(x). However, you can always swap out f(x) for Y.
• At Sal says that the definition is F(x) is going to be equal to x^2. Does that mean that if I have a function notation such as f(x)=x+4 and a given x is 2, do I have to square the 2?
• Yes - that is how it works, if you have f(x)=x² and are asked what is f(2), then you replace every instance of x in the function definition with 2 so given f(x) = x², that means f(2) = 2² = 4.
Here is another example: If f(x) = x² + 5x then f(2) = 2² + (5)(2) = 4 + 10 = 14
• Sorry if this was asked already but what is a non real number and what would an example be?
• Non-real numbers are called imaginary numbers and are based on i=√-1. You cannot take the square root of negative numbers, so you move to imaginary number.
• What if the input is an imaginary number (i) ? how will the graph be?
• what is parabola
• A parabola is created by a quadratic function such as y=x^2. It looks like a U.
• At why cant y be negative? Is this true for all problems or only dis one
• To answer you question of if it is always true, the answer is no. If you start from the quadratic parent function, y=x^2, then y cannot be negative.
One way to include negatives is to reflect it across the x axis by adding a negative y = -x^2. With this y cannot be positive and the range is y≤0. The other way to include negatives is to shift the function down. So y = x^2 -2 shifts the whole function down 2 units, and y ≥ -2.
• Umm, around Sal says, X can not be equal to 0. But if X was 0, wouldn't that be 0^2/0 = 0/0 = 1?

Or is there a rule that 0/0 doesn't equal 1? Is it undefined?
• Make a table for the function. Identify the range of the function
(1 vote)
• In Sal's last example, isn't it supposed to be g(x) instead of f(x) since he used a different variable in the first place?
• Yes he said the wrong thing but it should have been g(x)!

## Video transcript

- As a little bit of a review, we know that if we have some function, let's call it "f". We don't have to call it "f", but "f" is the letter most typically used for functions, that if I give it an input, a valid input, if I give it a valid input, and I use the variable "x" for that valid input, it is going to map that to an output. It is going to map that, or produce, given this x, it's going to product an output that we would call "f(x)." And we've already talked a little bit about the notion of a domain. A domain is the set of all of the inputs over which the function is defined. So if this the domain here, if this is the domain here, and I take a value here, and I put that in for x, then the function is going to output an f(x). If I take something that's outside of the domain, let me do that in a different color... If I take something that is outside of the domain and try to input it into this function, the function will say, "hey, wait wait," "I'm not defined for that thing" "that's outside of the domain." Now another interesting thing to think about, and that's actually what the focus of this video is, okay, we know the set of all of the valid inputs, that's called the domain, but what about all, the set of all of the outputs that the function could actually produce? And we have a name for that. That is called the range of the function. So the range. The range, and the most typical, there's actually a couple of definitions for range, but the most typical definition for range is "the set of all possible outputs." So you give me, you input something from the domain, it's going to output something, and by definition, because we have outputted it from this function, that thing is going to be in the range, and if we take the set of all of the things that the function could output, that is going to make up the range. So this right over here is the set of all possible, all possible outputs. All possible outputs. So let's make that a little bit more concrete, with an example. So let's say that I have the function f(x) defined as, so once again, I'm gonna input x's, and I have my function f, and I'm gonna output f(x). And let's say this def... The function definition here, the thing that tries to figure out, "okay, given an x, what f(x) do I produce?", the definition says "f(x) is going to be equal" "to whatever my input is, squared." Well, just as a little bit of review, we know what the domain here is going to be. The domain is the set of all valid inputs. So what are the valid inputs here? Well, I could take any real number and input into this, and I could take any real number and I can square it, there's nothing wrong with that, and so the domain is all real numbers. All, all real, all real numbers. But what's the range? Maybe I'll do that in a different color just to highlight it. What is going to be the range here, what is the set of all possible outputs? Well if you think about, actually, to help us think about, let me actually draw a graph here. Of what this looks like. What this looks like. So the graph of "f(x) is equal to x squared" is going to look something like this. So, it's gonna look, it's going to look something like this. I'm obviously hand-drawing it, so it's not perfect. It's gonna be a parabola with a, with a vertex right here at the origin. So this is the graph, this is the graph, "y is equal to f(x)," this of course is the x-axis, this of course is the y-axis. So let's think about it, what is the set of all possible outputs? Well in this case, the set of all possible outputs is the set of all possible y's here. Well, we see, y can take on any non-negative value. y could be zero, y could one, y could be pi, y could be e, but y cannot be negative. So the range here is, the range... We could, well we could say it a couple of ways, we could say, "f(x)", let me write it this way. "f(x) is a member of the real numbers" "such that, is such that f(x) is greater than" "or equal to zero." We could write it that way, if we wanted to write it in a less mathy notation, we could say that "f(x) is going to be" "greater than or equal to zero." f(x) is not going to be negative, so any non-negative number, the set of all non-negative numbers, that is our range. Let's do another example of this, just to make it a little bit, just to make it a little bit, a little bit clearer. Let's say that I had, let's say that I had g(x), let's say I have g(x), I'll do this in white, let's say it's equal to "x squared over x." So we could try to simplify g(x) a little bit, we could say, "look, if I have x squared" "and I divide it by x, that's gonna," "that's the same thing as g(x) being equal to x." "x squared over x" is x, but we have to be careful. Because right over here, we have to, in our domain, x cannot be equal to zero. If x is equal to zero, we get zero over zero, we get indeterminate form. So in order for this function to be the exact same function, we have to put that, 'cause it's not obvious now from the definition, we have to say, "x cannot be equal to zero." So g(x) is equal to x for any x as long as x is not equal to zero. Now these two function definitions are equivalent. And we could even graph it. We could graph it, it's going to look, I'm gonna do a quick and dirty version of this graph. It's gonna look something like, this. It's gonna have a slope of one, but it's gonna have a hole right at zero, 'cause it's not defined at zero. So it's gonna look like this. So the domain here, the domain of g is going to be, "x is a member of the real numbers" "such that x does not equal zero," and the range is actually going to be the same thing. The range here is going to be, we could say "f(x) is a member of the real numbers" "such that f(x) does not equal zero." "f(x) does not equal zero." So the domain is all real numbers except for zero, the range is all real numbers except for zero. So the big takeaway here is the range is all the pos... The set of all possible outputs of your function. The domain is the set of all valid inputs into your function.