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:

Which of the following is true of the function𝑓(𝑥)=𝑥1, where 𝑥0?

  • 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 .
  • BThe function is constant on .
  • CThe function is increasing on (,0].
  • DThe function is increasing on .
  • 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)=55𝑥 and accurately describes its monotonicity?

  • AThe domain is {1}, the range is {1}, and the function is decreasing over its domain.
  • BThe domain is {1}, the range is {1}, and the function is increasing over its domain.
  • CThe domain is , the range is , and the function is increasing over its domain.
  • DThe domain is , the range is , and the function is decreasing over its domain.
  • EThe 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)=(𝑥12)?

  • 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=𝑡68𝑡+63/𝑡0.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)=𝑥8𝑥 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 (𝑥)=17𝑥5?

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

Q14:

Which of the following statements is true for the function 𝑓(𝑥)=𝑥1𝑥1?

  • A 𝑓 ( 𝑥 ) is decreasing on .
  • B 𝑓 ( 𝑥 ) is increasing on (,1) and (1,).
  • C 𝑓 ( 𝑥 ) is increasing on .
  • 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)=𝜋3𝑥 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.

  • 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)=(𝑥12)?

  • 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)=15𝑥?

  • 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)=(𝑥6)?

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