in interval form the domain is given by
[ 0 , + infinity)
Example 1: Graph
f( x ) = SQRT (x)
and find the range of f.
Solution to Example 1:
Because the domain of f is the set of all positive real numbers and zero, we might construct a table of values as follows:
| x | 0 | 1 | 4 | 9 | 16 |
| SQRT (x) | 0 | 1 | 2 |
3 |
4 |
The values of x were selected so that the square root of these values are whole numbers which make it easy to plot the points shown in the table.
The range of f is given by the interval [0 , +infinity).
Example 2: Graph
f( x ) = SQRT (x - 3)
and find the range of f.
Solution to Example 2:
First find the domain of the square root function given above by stating that the expression under the square root must be positive or equal to zero
x - 3 ≥ 0
Solve the above inequality to obtain the domain of f as the set of all real values such that
x ≥ 3
We now select values of x in the domain to construct a table of values.
| x | 3 | 4 | 7 | 12 |
| SQRT (x - 3) | 0 | 1 | 2 |
3 |
The interval [0 , +infinity) represents the range of f.
Example 3: Graph
f( x ) = - SQRT (- 2x + 4) + 1
and find the range of f.
Solution to Example 3:
The domain of the function given above is found by setting
- 2x + 4 ≥ 0
Solve the above inequality to obtain the domain of f as the set of all real values such that
x ≤ 2
We now select values of x in the domain of f to construct a table of values. These values are selected so that the square root term is a whole number and give points that are easy to plot.
| x | 2 | 3/2 | 0 | -5/2 | -6 |
| - SQRT (-2 x + 4 ) + 1 | 1 | 0 | -1 |
-2 |
-3 |
The range of f is given by the interval (-infinity , 1].
Example 4: Graph
f( x ) = SQRT (- x 2 + 4)
and find the range of f.
Solution to Example 4:
The domain of function given above is found by solving the polynomial inequality
- x 2 + 4 ≥ 0
The solution set of the above inequality is given by the interval
[-2 , 2]
which is also the domain of the above function.
Let us write the given function as an equation as follows
y = SQRT (- x 2 + 4)
Square both sides and arrange to obtain.
x 2 + y 2 = 2 2
The equation obtained is that of a circle. Hence the graph of f(x) = SQRT (- x 2 + 4) is the upper half of a circle sinsce SQRT (- x 2 + 4) is positive. Hence the graph below.
The interval [0 , 2] represents the range of f.
Example 5: Graph
f( x ) = SQRT (x 2 - 9)
and find the range of f.
Solution to Example 5:
The domain of the function given above is found by solving
x 2 - 9 ≥ 0
Which gives a domain reprsented by
(-infinity , -3] U [3 , + infinity)
We now select values of x in the domain of f to construct a table of values, noting f(x) = f(-x) hence a symmetry of the graph with respect to the y axis.
| x | 3 | 5 | 8 |
| SQRT (x 2 - 9) | 0 | 4 | 7.4 |
The range of f is given by the interval [0 , + infinity).
Example 6: Graph
f( x ) = SQRT (x 2 - 6x + 9)
and find the range of f.
Solution to Example 6:
Let us use write the expression under the square root as a square as follows
x 2 - 6x + 9 = (x - 3) 2
Hence
f( x ) = SQRT (x 2 - 6x + 9)
= SQRT ( (x - 3) 2 ) = | x - 3 |
The given function has been rewitten as an absolute value function. Function f may be written as a piecewise function and graphed as follows.
The range of f is given by the interval [0 , + infinity).
Example 7: Graph
f( x ) = SQRT (x 2 + 4x + 6)
and find the range of f.
Solution to Example 7:
Use completing the square to rewtite the expression under the square root as follows
x 2 + 4x + 6 = (x + 2) 2 + 2
The expression under the square root is always positive hence the domain of f is the set of all real numbers. Let us first look at the graph of (x + 2) 2 + 2. It is a parabola.
We would expect the graph of f to have the same axis of symmetry, the vertical line, x = -2 as the above graph. The table of values may constructed as follows.
| x | -2 | 0 | 2 | 4 |
| SQRT ( (x + 2) 2 + 2 ) | 1.4 | 2.4 | 4.2 |
6.2 |
The range of f is given by the interval [SQRT(2) , + infinity).
More references and links on graphing.
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