Cách tìm Range

The domain of a function is the complete set of possible values of the independent variable.

In plain English, this definition means:

The domain is the set of all possible x-values which will make the function "work", and will output real y-values.

When finding the domain, remember:

• The denominator [bottom] of a fraction cannot be zero
• The number under a square root sign must be positive in this section

Example 1a

Here is the graph of `y = sqrt[x+4]`:

1 2 3 4 5 -1 -2 -3 -4 1 2 3 x y

Domain: `x>=-4`

The domain of this function is `x ≥ −4`, since x cannot be less than ` −4`. To see why, try out some numbers less than `−4` [like ` −5` or ` −10`] and some more than `−4` [like ` −2` or `8`] in your calculator. The only ones that "work" and give us an answer are the ones greater than or equal to ` −4`. This will make the number under the square root positive.

Notes:

1. The enclosed [colored-in] circle on the point `[-4, 0]`. This indicates that the domain "starts" at this point.
2. We saw how to draw similar graphs in section 4, Graph of a Function. For a more advanced discussion, see also How to draw y^2 = x − 2.

How to find the domain

In general, we determine the domain of each function by looking for those values of the independent variable [usually x] which we are allowed to use. [Usually we have to avoid 0 on the bottom of a fraction, or negative values under the square root sign].

The range of a function is the complete set of all possible resulting values of the dependent variable [y, usually], after we have substituted the domain.

In plain English, the definition means:

The range is the resulting y-values we get after substituting all the possible x-values.

How to find the range

• The range of a function is the spread of possible y-values [minimum y-value to maximum y-value]
• Substitute different x-values into the expression for y to see what is happening. [Ask yourself: Is y always positive? Always negative? Or maybe not equal to certain values?]
• Make sure you look for minimum and maximum values of y.
• Draw a sketch! In math, it's very true that a picture is worth a thousand words.

Example 1b

Let's return to the example above, `y = sqrt[x + 4]`.

We notice the curve is either on or above the horizontal axis. No matter what value of x we try, we will always get a zero or positive value of y. We say the range in this case is y ≥ 0.

1 2 3 4 5 -1 -2 -3 -4 1 2 3 x y

Range: `y>=0`

The curve goes on forever vertically, beyond what is shown on the graph, so the range is all non-negative values of `y`.

Example 2

The graph of the curve y = sin x shows the range to be betweeen −1 and 1.

1 2 3 4 5 -1 -2 -3 -4 -5 -6 -7 1 -1 x y

Range: `-1 -2`. The function on this side is defined as

`y = 3x + 2`

As x approaches `-2` from the right, we see that the function value also approaches `-4`.

The function is not defined at `x = -2` so it is not continuous there. We represent this with an open circle on the graph.

1 2 3 4 5 -1 -2 -3 -4 -5 5 10 15 -5 x y [-2, -4]

`y = -2x-8`

`y = 3x + 2`

Graph of a split function.

Example 4

`f[x]` `={[sin\ x,text[for ], x