Worksheet: Graphing Functions with Rational Exponents

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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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