Lesson 2-3: Lesson 2-3: Piecewise and Piecewise and Absolute Value Absolute Value

FunctionsFunctionsAdvanced Math TopicsAdvanced Math Topics

Piecewise functionsPiecewise functions A function defined by two or more equationsA function defined by two or more equations

Each equation applies to a different part of the domainEach equation applies to a different part of the domain

DomainDomain it is usually described in the problem and can it is usually described in the problem and can

just be rewritten for the answerjust be rewritten for the answer RangeRange

found by taking the endpoints (if there are any) found by taking the endpoints (if there are any) for the domain intervals and plugging them into for the domain intervals and plugging them into the appropriate function to find the y-valuesthe appropriate function to find the y-values

SummarySummary

f(x)= different f(x)= different things for different things for different sections of the sections of the graphgraph

Remember what Remember what open dots and open dots and closed dots mean??closed dots mean??

Piece Wise FunctionsPiece Wise Functions

Step 1 Graph the linear function f(x) = –1 for x ≤ 3. Since 3 satisfies this inequality, begin with a closed circle at (3, 2).

Step 2 Graph the constant function f(x) = –1 for x > 3. Since x does not satisfy this inequality, begin with an open circle at (3, –1) and draw a horizontal ray to the right.

Answer:Answer: The The function is defined function is defined for all values of for all values of xx, , so the domain is all so the domain is all real numbers. The real numbers. The values that are values that are yy--coordinates of coordinates of points on the graph points on the graph are all real are all real numbers less than numbers less than or equal to 2, so the or equal to 2, so the range is {range is {y y | | y y ≤ 2}≤ 2}..

A.A.

B.B.

C.C.

D.D.

Absolute Value FunctionAbsolute Value Function A function with a variable inside the A function with a variable inside the

absolute value barsabsolute value bars The graph makes a “V” shapeThe graph makes a “V” shape

To GraphTo Graph

Take everything inside the | | and set it Take everything inside the | | and set it equal to 0. The x-value you get is the x-equal to 0. The x-value you get is the x-value for the corner.value for the corner.

Plug the x-value in and solve for yPlug the x-value in and solve for y The resulting ordered pair is where you The resulting ordered pair is where you

absolute value graph starts! (the corner)absolute value graph starts! (the corner) Plug in two x-values on either side of Plug in two x-values on either side of

the corner to find ordered pairs to finish the corner to find ordered pairs to finish the graphthe graph

To Find Domain and To Find Domain and RangeRange

Domain: ARN there are no Domain: ARN there are no restrictions on the x-values you can restrictions on the x-values you can plug into an absolute value graphplug into an absolute value graph

Range: determined from the y-Range: determined from the y-values, y will be values, y will be >> or or << the y-value the y-value of the cornerof the corner

SummarySummary

x

xx

0

0

x

x

Absolute ValueAbsolute Value

GraphsGraphs

Graph Graph ff((xx) = |) = |xx| – 2 and | – 2 and gg((xx) = |) = |xx| | + 3 on the same coordinate + 3 on the same coordinate plane. Which answer choice is plane. Which answer choice is not true about the pair of not true about the pair of graphs?graphs?

Identify the type of Identify the type of functionfunction

Identify the type of Identify the type of functionfunction

Identify the type of Identify the type of functionfunction