Worksheet: Graphing Functions with Rational Exponents

In this worksheet, we will practice graphing functions with rational exponents then analyzing the function's domain, range, y-intercept, end behavior, continuity, and finally its decreasing and increasing intervals.

Q1:

The figure shows the graph of 𝑓 ( 𝑥 ) = 𝑥 .

Consider the behavior of the function 𝑔 ( 𝑥 ) = ( 𝑥 + 2 ) + 5 .

State the domain and the range of 𝑔 .

  • ADomain: ( , 2 ) ( 2 , ) , range: ( 5 , )
  • BDomain: ( , 2 ) ( 2 , ) , range: ( 5 , )
  • CDomain: ( , 5 ) ( 5 , ) , range: ( 2 , )
  • DDomain: ( , 2 ) ( 2 , ) , range: ( 5 , )
  • EDomain: ( , 5 ) ( 5 , ) , range: ( 2 , )

Find l i m 𝑔 ( 𝑥 ) .

Find l i m 𝑔 ( 𝑥 ) .

State where the function has a discontinuity.

  • A 𝑥 = 2
  • B 𝑥 = 5
  • C 𝑥 = 2
  • D 𝑥 = 5

State the intervals over which 𝑔 is increasing and decreasing.

  • AIncreasing over ( , 5 ) , decreasing over ( 5 , )
  • BIncreasing over ( , 2 ) , decreasing over ( 2 , )
  • CIncreasing over ( , 5 ) , decreasing over ( 5 , )
  • DIncreasing over ( , 2 ) , decreasing over ( 2 , )
  • EIncreasing over ( 2 , ) , decreasing over ( , 2 )

Q2:

Consider the function 𝑓 ( 𝑥 ) = 4 𝑥 .

State the domain and range of the function.

  • ADomain: ( , 0 ) , range: [ 0 , )
  • BDomain: ( , ) , range: ( 0 , )
  • CDomain: ( , 0 ) , range: ( 0 , )
  • DDomain: ( , ) , range: [ 0 , )
  • EDomain: ( , ) , range: ( , )

Find l i m 𝑓 ( 𝑥 ) .

  • A
  • B
  • C0

Find l i m 𝑓 ( 𝑥 ) .

  • A0
  • B
  • C

State the intervals over which the function is continuous.

  • A ( , )
  • B ( , 0 ]
  • C [ 0 , )
  • D ( , 0 )
  • E ( 0 , )

State the intervals over which the function is increasing and decreasing.

  • ADecreasing on [ 0 , ) , increasing on ( , 0 ]
  • BDecreasing on ( , 0 ] , increasing on [ 0 , )
  • CDecreasing on ( 0 , ) , increasing on ( , 0 )
  • DDecreasing on ( , 0 ) , increasing on ( 0 , )

Q3:

Consider the function .

State the domain and range of the function.

  • ADomain: , range:
  • BDomain: , range:
  • CDomain: , range:
  • DDomain: , range:
  • EDomain: , range:

Evaluate .

  • A
  • B0
  • C

State the interval over which the function is continuous.

  • A
  • B
  • C
  • D
  • E

State the intervals over which the function is increasing and/or decreasing.

  • AIncreasing on
  • BIncreasing on , decreasing on
  • CIncreasing on
  • DIncreasing on
  • EDecreasing on

Q4:

Consider the function .

State the domain and range of the function.

  • ADomain: , range:
  • BDomain: , range:
  • CDomain: , range:
  • DDomain: , range:
  • EDomain: , range:

Evaluate .

State the interval over which the function is continuous.

  • A
  • B
  • C
  • D
  • E

State the intervals over which the function is increasing and/or decreasing.

  • ADecreasing on
  • BIncreasing on
  • CIncreasing on , decreasing on
  • DDecreasing on
  • EDecreasing on , increasing on

Q5:

Consider the function .

State the domain and range of the function.

  • ADomain: , range:
  • BDomain: , range:
  • CDomain: , range:
  • DDomain: , range:
  • EDomain: , range:

Find .

Find .

  • A
  • B
  • C0

State where the function has a discontinuity.

  • A
  • B
  • C

State the intervals over which the function is increasing and decreasing.

  • AIncreasing over , decreasing over
  • BIncreasing over , decreasing over
  • CIncreasing over , decreasing over
  • DIncreasing over , decreasing over
  • EIncreasing over and , never decreasing

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