How To Find The Domain And Range of A Function Using It's Rule

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Drealup

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•     • Posts: 336 How To Find The Domain And Range of A Function Using It's Rule
Posted on: February 17, 2018, 10:10:29 PM

One may be confused when given a function rule to get it's range from it. It's very simple as far as you follow carefully.

Consider the function

f(x) =   5
____
x-3
.

Notice that there is no output number when the input number is 3 , since division by zero is meaningless.
Quote File image of an elementary function

This means x can have any value except 3. We say that 3 is not in the domain of the function f.
The domain of a function contains every possible input number unless otherwise specified.

The otherwise could be example, one could be given   g(x) = 3x + 7 for x > 2. Then the domain is the interval (2,∞). If it were not for the restriction on the domain, it would have a domain of (-∞,∞).

The range of a function

We call the set of all inputs of a function its domain, and we call the set of all outputs of a function its range.

It is much easier, in general, to look at the equation of a function and figure out its domain than it is to figure out its range.
For example, if

f(x) =  x + 2
______
x - 3.

We can see that its domain is all real numbers except 3.  And {x=R,x≠3} in interval notation.

However, it is not as easy to see what the the range must be. A good technique to use is to replace the f(x) in the equation with y and solve the equation for x(make x the subject). It would look this way:

x = 3y + 2
____
y - 1

Thus one can see that the output number can be anything except 1. Thus, the range of the function is all real numbers except 1. Range = {y:y=R,y≠1}

Goodluck!

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