piecewise functions & step functions

8/7/2019 Piecewise Functions & Step Functions

1/7

Grade Level:HG1

Course: Pre calculus

Branch: Functions

Topic: Piecewise Functions & Step Functions

Piecewise Functions

Consider the function f(x) graphed below. To describe this function, differentequations are needed for different parts of the domain of f. Therefore f is said tobe piecewise defined.

When x is less than 5, f(x) is linear with slope -2 and y-intercept 2. So,f(x) = -2x + 2 whenever x is less than 5.

When x is greater than or equal to 5, f(x) = 3.

f(x) is defined for all real numbers, so the domain of this function is all realnumbers.f(x) can never be less than or equal to -3, so the range of this function is all realnumbers greater than -3.


2/7

Example 1:

f( x ) = - x if x 2= x if x > 2

SolutionIn the above function, x 2, the formula for the function is f( x ) = -x and , if x >2 the formula is f( x ) = x. The domain of function fdefined above is the set of all

the real numbers since f is defined everywhere for all real numbers.

Example 2:f( x ) = 2 if x > -3

= -5 if x < -3SolutionThe above function is constant and equal to 2, if x>-3 .Function f is also constant and equal to -5 if x 4Solution:The above function is defined for all real numbers except for values of x in theinterval (-2, 2] and x = 4.


3/7

Example 5:f is a function defined byf( x ) = x 2 + 1 if x < 2

= - x + 3 if x 2Find the domain and range of function f and graph it.Solution:The domain of f is the set of all real numbers since function f is defined for all

real values of x.

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

In the interval [2, + ) the graph is a line with an x intercept at (3, 0) and passesthrough the point (2, 1). The interval [2, + ) is closed at x = 2 and the graphshows a "closed point". From the graph, we can observe that function f can take

all real values. The range is given by (- , + ).

Example 6:

f( x ) =1

x

if x < 0

= e -x if x 0Find the domain and range of function f and graph it.Solution:The domain of f is the set of all real numbers since function f is defined for all

real values of x.In the interval (- ,0) the graph of f is a hyperbola with vertical asymptote at x= 0.In the interval [0, + ) the graph is a decreasing exponential and passes throughthe point (0, 1). The interval [0,+ ) is closed at x = 0 and the graph shows a"closed point".


4/7

As x becomes very small,1

x

approaches zero. As x becomes very large, e -x also

approaches zero.

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

Example 7:Graph of the Piecewise Function y = -x + 3 on the interval [-3, 0]and y = 3x + 1 on the interval [0, 3]


5/7

Example 8:Graph of the piecewise function y = 2x + 3 on the interval (-3, 1)and y = 5 on the interval (1, 5)

Step function graphs

F(x) = [ x ]

The graphs look like a set of stairs, called step functions. These functions arediscrete, or not continuous.

Notice the endpoints in each step. Since these graphs are functions, there must

exist only one dependent-value (y) for every independent-integer (x). Therefore,one endpoint of a step is included (or closed) in the step with a black dot, and theother endpoint of the step is not included in the step (or open).


6/7

Example 1

A wholesale t-shirt manufacturer charges the following prices for t-shirt orders:$20 per shirt for shirt orders up to 20 shirts.$15 per shirt for shirt between 21 and 40 shirts.$10 per shirt for shirt orders between 41 and 80 shirts.

$5 per shirt for shirt orders over 80 shirts.

Sketch a graph of this discontinuous function.You've ordered 40 shirts and must pay shipping fees of $10. How much is yourtotal order?

Solution:

If I ordered 40 shirts and must pay $10 in shipping fees, then my total order will

cost $610. (40 * $15) +10 = 610.


7/7

QUIZ1 .

2. What is the domain of the above graph g(x)A) (- , 0]B) (- , )C) No real DomainD) None of the above

3. Giveng (x) = 3x + 1 for x < 0

= 1 for 0 x 5= x + 2 for x > 5

Find the value of g(-2)A) -5B) 7C) 1D) 0

piecewise functions & step functions

Documents