Graph The Piecewise Defined Function

A step by step tutorial on graphing and sketching piecewise functions. The graph, domain and range of these functions and other properties are examined. Free graph paper is available.

Definition of Piecewise Functions

A piecewise function is usually divers by more than one formula: a fomula for each interval.

Instance 1:

f( ten ) = - 10 if 10 <= 2

= x if 10 > 2

What the above says is that if x is smaller than or equal to two, the formula for the function is f( x ) = -x and if x is greater than ii, the formula is f( x ) = 10. It is as well of import to note that the domain of function f defined above is the ready of all the real numbers since f is divers everywhere for all existent numbers.

Example 2:

f( x ) = 2 if x > -iii

= -five if ten < -3

The higher up function is constant and equal to 2 if x is greater than -three. role f is besides constant and equal to -5 if 10 is less than -iii. Information technology tin can be said that function f is piecewise constant. The domain of f given above is the prepare of all existent numbers except -3: if x = -3 part f is undefined.

Example 3:

Functions involving absolute value are also a skillful case of piecewise functions.

f( ten ) = | ten |

Using the definition of the accented value, role f given higher up can be written

f( 10 ) = x if 10 >= 0

= -x if x < 0

The domain of the above function is the set of all existent numbers.

Instance iv:

Some other instance involving accented vaule.

f( x ) = | x + half-dozen |

The above function may be written every bit

f( x ) = x + half dozen if ten >= -6

= - (x + 6) if x < -6

The above function is defined for all existent numbers.

Example 5:

Some other example involving more than two intervals.

f( 10 ) = xⁱⁱ - 3 if x <= -10

= - 2x + 1 if -10 < x <= -2

= - x

³ if 2 < 10 < four

= ln x if 10 > 4

The above function is defined for all real numbers except for values of x in the interval (-2 , 2] and x = 4.

Example half dozen:

f is a function defined by

f( x ) = -1 if x <= -2

= 2 if x > -2

Find the domain and range of function f and graph it.

Solution to Example 6:

Function f is divers for all real values of x. The domain of f is the set of all real numbers. We will graph it past considering the value of the office in each interval.

In the interval (- inf , -two] the graph of f is a horizontal line y = f(x) = -1 (see formula for this interval above). Likewise this interval is closed at ten = -2 and therefore the graph must testify this : run into the "airtight bespeak" on the graph at x = -ii.

In the interval (-2 , + inf) the graph is a horizontal line y = f(ten) = 2 (come across formula for this interval higher up). The interval (-2 , + inf) is open at ten = -2 and the graph shows this with an "open point". Office f tin take only two values: -1 and ii. The range is given by {-1, 2}

graph of function in example 6

Instance vii:

f is a role divers past

f( x ) = 10² + i if ten < ii

= - x + iii if x >= two

Find the domain and range of function f and graph it.

Solution to Instance 7:

The domain of f is the set of all real numbers since function f is divers for all real values of x.

In the interval (- inf , 2) the graph of f is a parabola shifted up 1 unit. Also this interval is open at x = two and therefore the graph shows an "open up point" on the graph at ten = 2.

In the interval [two , + inf) the graph is a line with an x intercept at (iii , 0) and passes through the indicate (2 , 1). The interval [ii , + inf) is closed at x = two and the graph shows a "closed point". From the graph, nosotros can notice that function f can accept all real values. The range is given by (- inf, + inf).

graph of function in example 7

Example 8:

f is a function defined by

f( x ) = 1 / x if x < 0

= e^-10 if ten >= 0

Detect the domain and range of office f and graph information technology.

Solution to Example 8:

The domain of f is the set of all existent numbers since function f is defined for all real values of x.

In the interval (- inf , 0) the graph of f is a hyperbola with vertical asymptote at x = 0.

In the interval [0 , + inf) the graph is a decreasing exponential and passes through the point (0 , 1). The interval [0 , + inf) is closed at x = 0 and the graph shows a "airtight point".

As x becomes very small, 1 / x approaches zero. Every bit x becomes very large, e^-x besides approaches null. Hence the line y = 0 is a horizontal asymptote to the graph of f.

From the graph of f shown below, we can observe that function f can accept all existent values on (- inf , 0) U (0 , 1] which is the range of function f.

graph of function in example 8

Case 9:

f is a part defined past

f( x ) = -one if 10 <= -ane

= 1 if -ane < 10 <= 1

= x if 10 > 1

Detect the domain and range of office f and graph information technology.

Solution to Instance 9: The domain of f is the set of all real numbers.

In the interval (- inf , -i], the graph of f is a horizontal line y = f(10) = -i. Airtight bespeak at x = -1 since interval closed at x = -1.

In the interval (-1 , one] the graph is a horizontal line. There should a closed point at x = 1 merely read below.

In the interval (1 , + inf) the graph is the line y = x. There should an open point at 10 = 1 since the interval is open at ten = 1. But a airtight point (see higher up) and an open point at the same location becomes a "normal" signal.

From the graph of f shown beneath, we can find that function f tin can take all real values on {-1} U [i , + inf) which is the range of function f.