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:

Which of the following is true of the function𝑓(𝑥)=𝑥−1, where 𝑥≤0?

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

Q2:

What condition must there be on 𝑧 for 𝑓(𝑥)=𝑧7,where 𝑥 is a positive number, to be an increasing function?

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

Q3:

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

  • AThe function is decreasing on ℝ.
  • BThe function is constant on ℝ.
  • CThe function is increasing on (−∞,0].
  • DThe function is increasing on ℝ.
  • EThe function is constant on (−∞,0].

Q4:

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

  • AThe function is increasing on (−∞,−2) and constant on (−2,∞).
  • BThe function is decreasing on (9,∞) and constant on (−∞,9).
  • CThe function is decreasing on (−∞,−2) and constant on (−2,∞).
  • DThe function is increasing on (9,∞) and constant on (−∞,9).

Q5:

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

  • AThe domain is ℝ−{−1}, the range is {−1}, and the function is decreasing over its domain.
  • BThe domain is ℝ−{−1}, the range is {−1}, and the function is increasing over its domain.
  • CThe domain is ℝ, the range is ℝ, and the function is increasing over its domain.
  • DThe domain is ℝ, the range is ℝ, and the function is decreasing over its domain.
  • EThe 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−12)?

  • A 𝑓 ( 𝑥 ) is increasing on (−∞,12) and decreasing on (12,∞).
  • B 𝑓 ( 𝑥 ) is increasing on (12,∞) and decreasing on (−∞,12).
  • C 𝑓 ( 𝑥 ) is increasing on (−∞,−12) and decreasing on (−12,∞).
  • D 𝑓 ( 𝑥 ) is increasing on (−12,∞) and decreasing on (−∞,−12).

Q7:

A particle is moving in a straight line such that its velocity 𝑣 after 𝑡 seconds is given by 𝑣=−𝑡−68𝑡+63/𝑡≥0.ms, When is the velocity of the particle increasing?

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

Q8:

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

  • Aincreasing
  • Bconstant
  • 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 (−∞,0) and increasing on (0,∞).
  • BThe function is decreasing on (−∞,−5) and decreasing on (−5,∞).
  • CThe function is increasing on (−∞,−5) and increasing on (−5,∞).
  • DThe function is decreasing on (−∞,0) and decreasing on (0,∞).

Q10:

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

  • AThe function is increasing on the intervals (−5,−3) and (−3,∞) and decreasing on the interval (−∞,−3).
  • BThe function is increasing on the intervals (−∞,−5) and (−5,−3) and decreasing on the interval (−3,∞).
  • CThe function is increasing on the intervals (−∞,−5) and (−3,∞) and decreasing on the interval (−5,−3).
  • DThe function is increasing on the interval (−5,−3) and decreasing on the intervals (−∞,−5) and (−3,∞).
  • EThe function is increasing on the interval (−3,∞) and decreasing on the intervals (−∞,−5) and (−5,−3).

Q11:

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

  • AThe function is constant on (−∞,0] and constant on (0,∞).
  • BThe function is constant on (−∞,0] and constant on [−1,∞).
  • CThe function is constant on (−∞,1] and constant on (0,∞).
  • DThe function is constant on (−∞,1] and constant on [−1,∞).

Q12:

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

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

Q13:

Which of the following statements is true for the function ℎ(𝑥)=−17−𝑥−5?

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

Q14:

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

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

Q15:

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

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

Q16:

Determine the intervals of increase and decrease of the function 𝑓(𝑥)=−7𝑥 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 on the interval (0,∞) and decreasing on the interval (−∞,0).
  • D 𝑓 ( 𝑥 ) is increasing on the interval (−∞,−7) and decreasing on the interval (−7,∞).
  • E 𝑓 ( 𝑥 ) is increasing everywhere.

Q17:

Which of the following statements is true for the function 𝑓(𝑥)=−(𝑥−12)?

  • A 𝑓 ( 𝑥 ) is increasing on (−∞,12) and decreasing on (12,∞).
  • B 𝑓 ( 𝑥 ) is increasing on (12,∞) and decreasing on (−∞,12).
  • C 𝑓 ( 𝑥 ) is increasing on (−∞,−12) and decreasing on (−12,∞).
  • D 𝑓 ( 𝑥 ) is increasing on (−12,∞) and decreasing on (−∞,−12).

Q18:

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

  • AThe function is increasing on (5,8), constant on (−1,5), and decreasing on (−2,−1).
  • BThe function is increasing on (−2,−1), constant on (−1,5), and decreasing on (5,8).
  • CThe function is increasing on (5,8) and decreasing on (−2,5).
  • DThe function is increasing on (−2,5) 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?

  • AThe function is increasing on (−2,∞) and decreasing on (−∞,−2).
  • BThe function is increasing on (−∞,−1) and decreasing on (−1,∞).
  • CThe function is increasing on (−1,∞) and decreasing on (−∞,−1).
  • DThe 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 (2,∞) and decreasing on (−∞,2).
  • BThe function is increasing on (−∞,3) and decreasing on (−1,∞).
  • 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 decreasing on ℝ.
  • D 𝑓 ( 𝑥 ) is increasing on (−∞,−6) and decreasing on (−6,∞).

Q23:

Discuss the monotonicity of the following function on its domain.

  • AIncreasing on (−2,1)∪(4,∞) and decreasing on (−∞,−2)∪(1,4)
  • BIncreasing on (−∞,−2)∪(−2,1) and decreasing on (1,4)∪(4,∞)
  • CIncreasing on (1,4)∪(4,∞) and decreasing on (−∞,−2)∪(−2,1)
  • DIncreasing on (−∞,−2)∪(1,4) and decreasing on (−2,1)∪(4,∞)

Q24:

Which of the following statements is true for the function 𝑓(𝑥)=−10𝑥?

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

Q25:

Determine the intervals of increase and decrease of the function 𝑓(𝑥)=𝑥+10𝑥 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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