Worksheet: Increasing and Decreasing Intervals of a Function

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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 ℝ .
  • B The function is increasing on ℝ .
  • C The function is constant on ( − ∞ , 0 ] .
  • D The function is constant on ℝ .
  • 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 ℝ − { − 1 } , the range is { − 1 } , and the function is decreasing over its domain.
  • B The domain is ℝ , the range is ℝ , and the function is increasing over its domain.
  • C The domain is ℝ − { − 1 } , the range is { − 1 } , and the function is increasing over its domain.
  • D The domain is ℝ , the range is ℝ , and the function is decreasing over its domain.
  • E The domain is ℝ − { − 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 ℎ ( 𝑥 ) = − 1 7 − 𝑥 − 5 ?

  • A ℎ ( 𝑥 ) is decreasing on the intervals ( − ∞ , − 7 ) and ( − 7 , ∞ ) .
  • B ℎ ( 𝑥 ) is increasing on the intervals ( − ∞ , 7 ) and ( 7 , ∞ ) .
  • C ℎ ( 𝑥 ) is increasing on the intervals ( − ∞ , − 7 ) and ( − 7 , ∞ ) .
  • D ℎ ( 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 ℝ .
  • B 𝑓 ( 𝑥 ) is decreasing on ( − ∞ , 1 ) and ( 1 , ∞ ) .
  • C 𝑓 ( 𝑥 ) is decreasing on ℝ .
  • 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 ∈ ℝ .

  • 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 ℝ .
  • C 𝑓 ( 𝑥 ) is increasing on ( − 6 , ∞ ) and decreasing on ( − ∞ , − 6 ) .
  • D 𝑓 ( 𝑥 ) is decreasing on ℝ .

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 ℝ .
  • D 𝑓 ( 𝑥 ) is decreasing on ℝ .

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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