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### Course: Algebra 1>Unit 8

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 is parabola
• A parabola is created by a quadratic function such as y=x^2. It looks like a U.
• What if the input is an imaginary number (i) ? how will the graph be?
• 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.
• What is a parabola?