Section 5.2B Power amd Piecewise Defined Functions

Definition: A function is a function of the form kxr, where k and r are

real numbers.

Some Definitions Involving Exponents:

1. If a 6= 0, a0 = 1, a�1 = 1a , and, in general, a�x = 1

ax .

2. a1/2 =pa, a1/3 = 3

pa, and, in general, a1/n = n

pa.

3. am/n = npam = ( n

pa)m

The domain of f(x) = npg(x) is all real numbers (�1,1) if n is odd. If n is even then to find

the domain of f(x) = np

g(x), you must solve g(x) � 0 for x.

Example 1: Find the domain of the following power functions:

a) f(x) = 3px2 � 1

a) f(x) =px� 1

b)4x� 24px� 1

power

3 is odd, so domain is (-o@

X - 120 Domain+ I + I

X Z / [@

Root function vs on the bottom

of the fraction , so solve × -1 > 0 for × ( Can't have * Obooytoathe)

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b)

px

x2 � 3x� 4

Example 2: Rationalize the numerator of the function

px+ 2� 5

x� 2

Example 3: Compute the following for the function defined below. Simplify your answers.

f(x) =px+ 5

a) f(x+ h)

b) f(x+ h)� f(x)

2 Spring 2018, Maya Johnson

Both : xzo &X¥4&x¥ - 1

Top first : xzo Ignore -1 since it 's < 0.

BottomlyIYEXYIEO 894k€( × - 4) =o or ( xtD=o

×=4 or X= - 1

Multiply by th Conjugate -7 LEE

~M

( €2 +5)

Middle terms =(F+z - 5)(F+z + 5) =×f+2XFt2) - (5×5) .

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c)f(x+ h)� f(x)

h

Definition: Functions whose definition involve more than one rule are calledPiecewise Functions.

To graph, graph each rule over the appropriate portion of the domain.

Example 4, Absolute value function: The absolute value of a number x, denoted by |x|,is the distance from x to 0 on the real number line. The plot and definition of absolute value

function if given below.

x

y

|x| =(x x � 0

�x x < 0

Example 5: Write f(x) = |6x+ 7| as a piecewise defined function.

3 Spring 2018, Maya Johnson

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=

txnFsfxtsXxfhFtxF5LxFhsXFthtsHxFsXFFDhTxEsxiFs-E5xFsEIhEIFYIasjxEEtEsTiaatF@t6xt7l-6xt7when.t

720's

,16×+71 = - (6×+7) who ex +7<0

Solve for × in bolt.

ask '

6x±}>=q⇒ 6 ÷ .

.

-

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Therefore 6×+7<0 ⇒ X <- 7/6

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Example 6: Find the domain and sketch graph of the function

f(x) =

8<

:x+ 1 if x < 1

1� x if x > 1

Example 7: Evaluate f(�2), f(�1), and f(4) for the following piecewise defined function.

f(x) =

8<

:x+ 5 if x �1

x2 if x > �1

4 Spring 2018, Maya Johnson

× = I is not defined

So domain

istoeiDUl@1gff4.t

2) +5 =Dftp..tl ) +5=40

fl 4) = (4) 2=160

Sketch a graph of the function.

Example 8: Find the domain of the following piecewise defined function.

f(x) =

8<

:

2

x2 � 9x+ 14if 0 x 4

4x+ 5 if x > 4

Example 9: An electricity company charges its customers a base rate of 10 a month, plus 6

cents per kilowatt-hour(kWh) for the first 1200 kWh and 7 cents per kWh for all usage over 1200

kWh. Express the monthly cost E as a function of the amount x of electricity used.

5 Spring 2018, Maya Johnson

TV

for 2-so Domain T

5×2-9×+14EEAYXIIEEO any× = 7 or X=Z × = 4 rs also not a problem

but x=7 is not in the range of OE x<- 4

Cost : 10 +.

obx if 0 a- ×± Koo

But ,of × 7 1200 , say 1201 ,

the first 1200 costs

10 +. 064204=82 ,

and the 1kWh over is.

07

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