Worksheet: Increasing and Decreasing Intervals of a Function

In this worksheet, we will practice finding the intervals over which a function is increasing, constant, or decreasing.

Q1:

Discuss the monotony of the function , where .

  • A is increasing on the interval and decreasing on the interval .
  • B is increasing on the interval .
  • C is decreasing on the interval and increasing on the interval .
  • D is decreasing on the interval .
  • E is increasing on the interval .

Q2:

What condition must there be on 𝑧 for 𝑓 ( 𝑥 ) = 𝑧 7 𝑥 ,where 𝑥 is a positive number, to be an increasing function?

  • A 𝑧 > 0
  • B 𝑧 > 1
  • C 𝑧 < 7
  • D 𝑧 > 7

Q3:

The graph of a function is given below. Which of the following statements about the function is true?

  • A The function is decreasing on .
  • B The function is increasing on .
  • C The function is constant on ( , 0 ] .
  • D The function is constant on .
  • E The function is increasing on ( , 0 ] .

Q4:

Which of the following correctly describes the monotonicity of the function represented by the graph in the figure below?

  • A The function is increasing on ( 9 , ) and constant on ( , 9 ) .
  • B The function is decreasing on ( , 2 ) and constant on ( 2 , ) .
  • C The function is increasing on ( , 2 ) and constant on ( 2 , ) .
  • D The function is decreasing on ( 9 , ) and constant on ( , 9 ) .

Q5:

Which of the following statements correctly defines the domain and range of the function 𝑓 ( 𝑥 ) = 5 5 𝑥 and accurately describes its monotonicity?

  • A The domain is { 1 } , the range is { 1 } , and the function is decreasing over its domain.
  • B The domain is , the range is , and the function is increasing over its domain.
  • C The domain is { 1 } , the range is { 1 } , and the function is increasing over its domain.
  • D The domain is , the range is , and the function is decreasing over its domain.
  • E The domain is { 1 , 5 } , the range is { 1 , 5 } , and the function is decreasing over its domain.

Q6:

Which of the following statements is true for the function 𝑓 ( 𝑥 ) = ( 𝑥 1 2 ) ?

  • A 𝑓 ( 𝑥 ) is increasing on ( 1 2 , ) and decreasing on ( , 1 2 ) .
  • B 𝑓 ( 𝑥 ) is increasing on ( , 1 2 ) and decreasing on ( 1 2 , ) .
  • C 𝑓 ( 𝑥 ) is increasing on ( , 1 2 ) and decreasing on ( 1 2 , ) .
  • D 𝑓 ( 𝑥 ) is increasing on ( 1 2 , ) and decreasing on ( , 1 2 ) .

Q7:

A particle is moving in a straight line such that its velocity 𝑣 after 𝑡 seconds is given by 𝑣 = 𝑡 6 8 𝑡 + 6 3 / 𝑡 0 . m s , When is the velocity of the particle increasing?

  • A 𝑡 < 6 8 s
  • B 𝑡 > 3 4 s
  • C 𝑡 > 6 8 s
  • D 𝑡 < 3 4 s

Q8:

Is the function shown increasing, decreasing, or constant in the interval ( 2 , 2 ) ?

  • Aconstant
  • Bincreasing
  • Cdecreasing

Q9:

The graph of a function is given below. Which of the following statements about the function is true?

  • AThe function is increasing on ( , 5 ) and increasing on ( 5 , ) .
  • BThe function is decreasing on ( , 0 ) and decreasing on ( 0 , ) .
  • CThe function is decreasing on ( , 5 ) and decreasing on ( 5 , ) .
  • DThe function is increasing on ( , 0 ) and increasing on ( 0 , ) .

Q10:

Given that 𝑓 ( 𝑥 ) = 5 𝑥 + ( 𝑥 + 4 ) 5 5 , determine the intervals on which 𝑓 is increasing or decreasing.

  • AThe function is increasing on the intervals ] , 5 [ and ] 5 , 3 [ and decreasing on the interval ] 3 , [ .
  • BThe function is increasing on the interval ] 5 , 3 [ and decreasing on the intervals ] , 5 [ and ] 3 , [ .
  • CThe function is increasing on the interval ] 3 , [ and decreasing on the intervals ] , 5 [ and ] 5 , 3 [ .
  • D The function is increasing on the intervals ] , 5 [ and ] 3 , [ and decreasing on the interval ] 5 , 3 [ .
  • EThe function is increasing on the intervals ] 5 , 3 [ and ] 3 , [ and decreasing on the interval ] , 3 [ .

Q11:

The graph of a function is given below. Which of the following statements about the function is true?

  • A The function is constant on ( , 0 ] and constant on [ 1 , ) .
  • B The function is constant on ( , 1 ] and constant on [ 1 , ) .
  • C The function is constant on ( , 1 ] and constant on ( 0 , ) .
  • D The function is constant on ( , 0 ] and constant on ( 0 , ) .

Q12:

Determine the intervals of increase and decrease of the function 𝑓 ( 𝑥 ) = 𝑥 8 𝑥 given that 𝑥 ( 8 , 1 ) .

  • A 𝑓 ( 𝑥 ) is increasing on the interval ( , 4 ) and decreasing on the interval ( 4 , ) .
  • B 𝑓 ( 𝑥 ) is increasing on the interval ( 4 , 1 ) and decreasing on the interval ( 8 , 4 ) .
  • C 𝑓 ( 𝑥 ) is increasing on the interval ( 4 , ) and decreasing on the interval ( , 4 ) .
  • D 𝑓 ( 𝑥 ) is increasing on the interval ( 8 , 4 ) and decreasing on the interval ( 4 , 1 ) .

Q13:

Which of the following statements is true for the function ( 𝑥 ) = 1 7 𝑥 5 ?

  • A ( 𝑥 ) is decreasing on the intervals ( , 7 ) and ( 7 , ) .
  • B ( 𝑥 ) is increasing on the intervals ( , 7 ) and ( 7 , ) .
  • C ( 𝑥 ) is increasing on the intervals ( , 7 ) and ( 7 , ) .
  • D ( 𝑥 ) is decreasing on the intervals ( , 7 ) and ( 7 , ) .

Q14:

Which of the following statements is true for the function 𝑓 ( 𝑥 ) = 𝑥 1 𝑥 1 ?

  • A 𝑓 ( 𝑥 ) is increasing on .
  • B 𝑓 ( 𝑥 ) is decreasing on ( , 1 ) and ( 1 , ) .
  • C 𝑓 ( 𝑥 ) is decreasing on .
  • D 𝑓 ( 𝑥 ) is increasing on ( , 1 ) and ( 1 , ) .
  • E 𝑓 ( 𝑥 ) is increasing on ( 1 , ) and decreasing on ( , 1 ) .

Q15:

How do the values of the function 𝑓 ( 𝑥 ) = 𝜋 3 𝑥 behave as 𝑥 increases?

  • AThe values of 𝑓 ( 𝑥 ) are constant.
  • BThe values of 𝑓 ( 𝑥 ) increase.
  • CThe values of 𝑓 ( 𝑥 ) decrease.

Q16:

Determine the intervals of increase and decrease of the function 𝑓 ( 𝑥 ) = 7 𝑥 given that 𝑥 .

  • A 𝑓 ( 𝑥 ) is increasing on the interval ( , 7 ) and decreasing on the interval ( 7 , ) .
  • B 𝑓 ( 𝑥 ) is increasing on the interval ( 0 , ) and decreasing on the interval ( , 0 ) .
  • C 𝑓 ( 𝑥 ) is increasing everywhere.
  • D 𝑓 ( 𝑥 ) is increasing on the interval ( , 0 ) and decreasing on the interval ( 0 , ) .
  • E 𝑓 ( 𝑥 ) is increasing on the interval ( 7 , ) and decreasing on the interval ( , 7 ) .

Q17:

Which of the following statements is true for the function 𝑓 ( 𝑥 ) = ( 𝑥 1 2 ) ?

  • A 𝑓 ( 𝑥 ) is increasing on ( , 1 2 ) and decreasing on ( 1 2 , ) .
  • B 𝑓 ( 𝑥 ) is increasing on ( 1 2 , ) and decreasing on ( , 1 2 ) .
  • C 𝑓 ( 𝑥 ) is increasing on ( 1 2 , ) and decreasing on ( , 1 2 ) .
  • D 𝑓 ( 𝑥 ) is increasing on ( , 1 2 ) and decreasing on ( 1 2 , ) .

Q18:

Which of the following statements correctly describe the monotony of the function represented in the figure below?

  • A The function is increasing on ( 2 , 5 ) and decreasing on ( 5 , 8 ) .
  • B The function is increasing on ( 5 , 8 ) , constant on ( 1 , 5 ) , and decreasing on ( 2 , 1 ) .
  • C The function is increasing on ( 5 , 8 ) and decreasing on ( 2 , 5 ) .
  • D The function is increasing on ( 2 , 1 ) , constant on ( 1 , 5 ) , and decreasing on ( 5 , 8 ) .

Q19:

Which of the following statements is true for the function 𝑓 ( 𝑥 ) = 1 5 𝑥 ?

  • A 𝑓 ( 𝑥 ) is increasing on ( , 5 ) and increasing on ( 5 , ) .
  • B 𝑓 ( 𝑥 ) is decreasing on ( , 5 ) and decreasing on ( 5 , ) .
  • C 𝑓 ( 𝑥 ) is decreasing on ( , 5 ) and decreasing on ( 5 , ) .
  • D 𝑓 ( 𝑥 ) is increasing on ( , 5 ) and increasing on ( 5 , ) .

Q20:

Which of the following statements correctly describe the monotony on the domain of the function represented in the figure below?

  • A The function is increasing on ( , 2 ) and decreasing on ( 2 , ) .
  • B The function is increasing on ( , 1 ) and decreasing on ( 1 , ) .
  • C The function is increasing on ( 1 , ) and decreasing on ( , 1 ) .
  • D The function is increasing on ( 2 , ) and decreasing on ( , 2 ) .

Q21:

The graph of a function is given below. Which of the following statements about the function is true?

  • AThe function is increasing on ( , 3 ) and decreasing on ( 1 , ) .
  • BThe function is increasing on ( , 2 ) and decreasing on ( 2 , ) .
  • CThe function is increasing on ( 1 , ) and decreasing on ( , 3 ) .
  • DThe function is increasing on ( 2 , ) and decreasing on ( , 2 ) .

Q22:

Which of the following statements is true for the function 𝑓 ( 𝑥 ) = ( 𝑥 6 ) ?

  • A 𝑓 ( 𝑥 ) is increasing on ( , 6 ) and decreasing on ( 6 , ) .
  • B 𝑓 ( 𝑥 ) is increasing on .
  • C 𝑓 ( 𝑥 ) is increasing on ( 6 , ) and decreasing on ( , 6 ) .
  • D 𝑓 ( 𝑥 ) is decreasing on .

Q23:

Discuss the monotonicity of the following function on its domain.

  • Aincreasing on ( 1 , 4 ) ( 4 , ) and decreasing on ( , 2 ) ( 2 , 1 )
  • Bincreasing on ( 2 , 1 ) ( 4 , ) and decreasing on ( , 2 ) ( 1 , 4 )
  • Cincreasing on ( , 2 ) ( 2 , 1 ) and decreasing on ( 1 , 4 ) ( 4 , )
  • Dincreasing on ( , 2 ) ( 1 , 4 ) and decreasing on ( 2 , 1 ) ( 4 , )

Q24:

Which of the following statements is true for the function 𝑓 ( 𝑥 ) = 1 0 𝑥 ?

  • A 𝑓 ( 𝑥 ) is increasing on ( , 0 ) and decreasing on ( 0 , ) .
  • B 𝑓 ( 𝑥 ) is increasing on ( 0 , ) and decreasing on ( , 0 ) .
  • C 𝑓 ( 𝑥 ) is increasing on .
  • D 𝑓 ( 𝑥 ) is decreasing on .

Q25:

Determine the intervals of increase and decrease of the function 𝑓 ( 𝑥 ) = 𝑥 + 1 0 𝑥 given that 𝑥 [ 7 , 1 ] .

  • A 𝑓 ( 𝑥 ) is increasing on the interval ( 5 , ) and decreasing on the interval ( , 5 ) .
  • B 𝑓 ( 𝑥 ) is increasing on the interval ( 7 , 5 ) and decreasing on the interval ( 5 , 1 ) .
  • C 𝑓 ( 𝑥 ) is increasing on the interval ( , 5 ) and decreasing on the interval ( 5 , ) .
  • D 𝑓 ( 𝑥 ) is increasing on the interval ( 5 , 1 ) and decreasing on the interval ( 7 , 5 ) .

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