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.

Graphing Functions