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Lesson: Sign of a Function

In this lesson, we will learn how to determine the sign of a constant, linear, or quadratic function from its equation.

Sample Question Videos

01:13

Worksheet • 25 Questions • 1 Video

Q1:

For which values of π‘₯ is the function 𝑓 ( π‘₯ ) = 8 π‘₯ βˆ’ 1 3 positive?

  • A π‘₯ > 1 3 8
  • B π‘₯ > βˆ’ 1 3 8
  • C π‘₯ < 1 3 8
  • D π‘₯ ≀ 1 3 8
  • E π‘₯ β‰₯ 1 3 8

Q2:

For which values of π‘₯ is the function 𝑓 ( π‘₯ ) = βˆ’ 5 π‘₯ + 2 negative?

  • A π‘₯ > 2 5
  • B π‘₯ > βˆ’ 2 5
  • C π‘₯ < 2 5
  • D π‘₯ ≀ 2 5
  • E π‘₯ β‰₯ 2 5

Q3:

Determine the sign of the function 𝑓 ( π‘₯ ) = βˆ’ 1 5 π‘₯ in ℝ .

  • AThe function is positive when π‘₯ < 0 , negative when π‘₯ > 0 , and equals zero when π‘₯ = 0 .
  • BThe function is positive when π‘₯ ∈ ℝ βˆ’ { 0 } , and it equals zero when π‘₯ = 0 .
  • CThe function is negative for all π‘₯ ∈ ℝ .
  • DThe function is positive when π‘₯ > 0 , negative when π‘₯ < 0 , and equals zero when π‘₯ = 0 .
  • EThe function is positive for all π‘₯ ∈ ℝ .

Q4:

Determine the sign of the function 𝑓 ( π‘₯ ) = 7 π‘₯ + 1 6 √ 7 π‘₯ + 6 4 2 in ℝ .

  • AThe function is positive when π‘₯ ∈ ℝ βˆ’  βˆ’ 8 √ 7  and is zero when π‘₯ = βˆ’ 8 √ 7 .
  • BThe function is positive when π‘₯ ∈ ℝ βˆ’  √ 7 8  and is zero when π‘₯ = √ 7 8 .
  • CThe function is negative when π‘₯ ∈ ℝ βˆ’  βˆ’ 8 √ 7  and is zero when π‘₯ = βˆ’ 8 √ 7 .
  • DThe function is negative when π‘₯ ∈ ℝ βˆ’  8 √ 7  and is zero when π‘₯ = 8 √ 7 .
  • EThe function is positive when π‘₯ ∈ ℝ βˆ’  8 √ 7  and is zero when π‘₯ = 8 √ 7 .

Q5:

Determine the sign of the function 𝑓 ( π‘₯ ) = 3 π‘₯ βˆ’ 2 √ 3 π‘₯ + 1 2 in ℝ .

  • AThe function is positive when π‘₯ ∈ ℝ βˆ’  1 √ 3  and is zero when π‘₯ = 1 √ 3 .
  • BThe function is positive when π‘₯ ∈ ℝ βˆ’  βˆ’ √ 3  and is zero when π‘₯ = βˆ’ √ 3 .
  • CThe function is negative when π‘₯ ∈ ℝ βˆ’  1 √ 3  and is zero when π‘₯ = 1 √ 3 .
  • DThe function is negative when π‘₯ ∈ ℝ βˆ’  βˆ’ 1 √ 3  and is zero when π‘₯ = βˆ’ 1 √ 3 .
  • EThe function is positive when π‘₯ ∈ ℝ βˆ’  βˆ’ 1 √ 3  and is zero when π‘₯ = βˆ’ 1 √ 3 .

Q6:

Determine the sign of the function 𝑓 ( π‘₯ ) = βˆ’ 5 π‘₯ + 5 .

  • A The function is positive when π‘₯ < 1 , the function is negative when π‘₯ > 1 , and the function equals zero when π‘₯ = 1 .
  • Bpositive
  • C The function is positive when π‘₯ > 1 , the function is negative when π‘₯ < 1 , and the function equals zero when π‘₯ = 1 .
  • D The function is positive when π‘₯ < βˆ’ 1 , the function is negative when π‘₯ > βˆ’ 1 , and the function equals zero when π‘₯ = βˆ’ 1 .
  • E The function is positive when π‘₯ > βˆ’ 1 , the function is negative when π‘₯ < βˆ’ 1 , and the function equals zero when π‘₯ = βˆ’ 1 .

Q7:

Determine the sign of the function 𝑓 ( π‘₯ ) = 2 .

  • AThe function is positive for all π‘₯ ∈ ℝ .
  • BThe function is positive when π‘₯ > 0 , negative when π‘₯ < 0 , and equals zero when π‘₯ = 0 .
  • CThe function is positive when π‘₯ ∈ ℝ βˆ’ { 0 } , and equals zero when π‘₯ = 0 .
  • DThe function is positive when π‘₯ < 0 , negative when π‘₯ > 0 , and equals zero when π‘₯ = 0 .

Q8:

Determine the sign of the function 𝑓 ( π‘₯ ) = βˆ’ 4 .

  • AThe function is negative for all π‘₯ ∈ ℝ .
  • BThe function is positive when π‘₯ > 0 , negative when π‘₯ < 0 , and equals zero when π‘₯ = 0 .
  • CThe function is positive when π‘₯ ∈ ℝ βˆ’ { 0 } , and equals zero when π‘₯ = 0 .
  • DThe function is positive when π‘₯ < 0 , negative when π‘₯ > 0 , and equals zero when π‘₯ = 0 .

Q9:

In which of the following intervals is 𝑓 ( π‘₯ ) = βˆ’ 8 negative?

  • A ] βˆ’ ∞ , ∞ [
  • B ] 8 , ∞ [
  • C ] βˆ’ 8 , 8 [
  • D ] βˆ’ 8 , ∞ [
  • E ] βˆ’ ∞ , 8 [

Q10:

In which of the following intervals is 𝑓 ( π‘₯ ) = 1 3 positive?

  • A ] βˆ’ ∞ , ∞ [
  • B ] 1 3 , ∞ [
  • C ] βˆ’ 1 3 , 1 3 [
  • D ] βˆ’ 1 3 , ∞ [
  • E ] βˆ’ ∞ , 1 3 [

Q11:

Which of the following statements about the sign of the function 𝑓 ( π‘₯ ) = ( π‘₯ + 3 ) 2 is correct?

  • A 𝑓 ( π‘₯ ) is positive for π‘₯ ∈ βˆ’ 3 .
  • B 𝑓 ( π‘₯ ) is positive for π‘₯ ∈ 9 .
  • C 𝑓 ( π‘₯ ) is positive for π‘₯ ∈ 3 .
  • D 𝑓 ( π‘₯ ) is negative for π‘₯ ∈ βˆ’ 3 .
  • E 𝑓 ( π‘₯ ) is negative for π‘₯ ∈ 3 .

Q12:

What is the condition on π‘₯ so that 𝑓 ( π‘₯ ) = 3 π‘₯ is negative?

  • A π‘₯ < 0
  • B π‘₯ ∈ ℝ
  • C π‘₯ ∈ ℝ βˆ’ { 0 }
  • D π‘₯ 𝑔 𝑑 ; 0

Q13:

What is the condition on π‘₯ so that 𝑓 ( π‘₯ ) = 1 1 π‘₯ is positive?

  • A π‘₯ 𝑔 𝑑 ; 0
  • B π‘₯ ∈ ℝ
  • C π‘₯ ∈ ℝ βˆ’ { 0 }
  • D π‘₯ < 0

Q14:

Determine where the function 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 4 0 0 2 is positive.

  • A ℝ βˆ’ [ βˆ’ 2 0 , 2 0 ]
  • B ] 2 0 , ∞ [
  • C ] βˆ’ 2 0 , 2 0 [
  • D ] βˆ’ ∞ , βˆ’ 2 0 [
  • E ] βˆ’ ∞ , 2 0 [

Q15:

Determine the sign of the function 𝑓 ( π‘₯ ) = π‘₯ + 1 0 π‘₯ + 1 6 2 .

  • A The function is positive when π‘₯ ∈ ℝ βˆ’ [ βˆ’ 8 , βˆ’ 2 ] , the function is negative when π‘₯ ∈ ] βˆ’ 8 , βˆ’ 2 [ , and the function equals zero when π‘₯ ∈ { βˆ’ 8 , βˆ’ 2 } .
  • B The function is positive when π‘₯ ∈ ] βˆ’ 8 , βˆ’ 2 [ , the function is negative when π‘₯ ∈ ℝ βˆ’ [ βˆ’ 8 , βˆ’ 2 ] , and the function equals zero when π‘₯ ∈ { βˆ’ 8 , βˆ’ 2 } .
  • C The function is positive when π‘₯ ∈ [ βˆ’ 8 , βˆ’ 2 ] , the function is negative when π‘₯ ∈ ℝ βˆ’ { βˆ’ 8 , βˆ’ 2 } , and the function equals zero when π‘₯ ∈ { βˆ’ 8 , βˆ’ 2 } .
  • D The function is positive when π‘₯ ∈ ℝ βˆ’ { βˆ’ 8 , βˆ’ 2 } , the function is negative when π‘₯ ∈ [ βˆ’ 8 , βˆ’ 2 ] , and the function equals zero when π‘₯ ∈ { βˆ’ 8 , βˆ’ 2 } .

Q16:

Determine the sign of the function 𝑓 ( π‘₯ ) = π‘₯ + 8 π‘₯ βˆ’ 9 2 .

  • A The function is positive when π‘₯ ∈ ℝ βˆ’ [ βˆ’ 9 , 1 ] , the function is negative when π‘₯ ∈ ] βˆ’ 9 , 1 [ , and the function equals zero when π‘₯ ∈ { βˆ’ 9 , 1 } .
  • B The function is positive when π‘₯ ∈ ] βˆ’ 9 , 1 [ , the function is negative when π‘₯ ∈ ℝ βˆ’ [ βˆ’ 9 , 1 ] , and the function equals zero when π‘₯ ∈ { βˆ’ 9 , 1 } .
  • C The function is positive when π‘₯ ∈ [ βˆ’ 9 , 1 ] , the function is negative when π‘₯ ∈ ℝ βˆ’ { βˆ’ 9 , 1 } , and the function equals zero when π‘₯ ∈ { βˆ’ 9 , 1 } .
  • D The function is positive when π‘₯ ∈ ℝ βˆ’ { βˆ’ 9 , 1 } , the function is negative when π‘₯ ∈ [ βˆ’ 9 , 1 ] , and the function equals zero when π‘₯ ∈ { βˆ’ 9 , 1 } .

Q17:

Determine the sign of the function 𝑓 ( π‘₯ ) = 5 βˆ’ 1 7 π‘₯ .

  • AThe function is positive when π‘₯ < 3 5 , negative when π‘₯ > 3 5 , and zero when π‘₯ = 3 5 .
  • BThe function is positive when π‘₯ ∈ ℝ βˆ’ { 5 } , and zero when π‘₯ = 5 .
  • CThe function is negative when π‘₯ < 3 5 , positive when π‘₯ > 3 5 , and zero when π‘₯ = 3 5 .
  • DThe function is negative when π‘₯ < βˆ’ 5 , positive when π‘₯ > βˆ’ 5 , and zero when π‘₯ = βˆ’ 5 .
  • EThe function is positive when π‘₯ < βˆ’ 3 5 , negative when π‘₯ > βˆ’ 3 5 , and zero when π‘₯ = βˆ’ 3 5 .

Q18:

What are the values of π‘₯ for which the functions 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 5 and 𝑔 ( π‘₯ ) = π‘₯ + 2 π‘₯ βˆ’ 4 8 2 are both positive?

  • A π‘₯ > 6
  • B π‘₯ > βˆ’ 8
  • C π‘₯ < 6
  • D π‘₯ > 5
  • E π‘₯ < βˆ’ 8

Q19:

The function 𝑓 ( π‘₯ ) = βˆ’ ( π‘₯ βˆ’ 1 ) ( π‘₯ + 6 ) is positive in which interval?

  • A ] βˆ’ 6 , 1 [
  • B ℝ βˆ’ [ βˆ’ 6 , 1 ]
  • C [ βˆ’ 1 , 6 ]
  • D ℝ βˆ’ ] βˆ’ 1 , 6 [
  • E ] βˆ’ 1 , 6 [

Q20:

Determine the interval where the sign of each of the two functions 𝑓 ( π‘₯ ) = 2 π‘₯ βˆ’ 7 π‘₯ βˆ’ 3 0 2 and 𝑔 ( π‘₯ ) = π‘₯ βˆ’ 3 π‘₯ βˆ’ 1 0 2 are negative together on ℝ .

  • A ] βˆ’ 2 , 5 [
  • B  βˆ’ ∞ , βˆ’ 5 2 
  • C ℝ βˆ’ [ βˆ’ 2 , 5 ]
  • D ] 6 , ∞ [
  • E  βˆ’ ∞ , βˆ’ 5 2 [ βˆͺ ] 6 , ∞ 

Q21:

Determine the sign of the function 𝑓 ( π‘₯ ) = 4 βˆ’ 2 5 π‘₯ 2 .

  • AThe function is positive at π‘₯ ∈ ο€Ό βˆ’ 2 5 , 2 5  , negative at π‘₯ ∈ ℝ βˆ’  βˆ’ 2 5 , 2 5  , and equals zero when π‘₯ ∈  βˆ’ 2 5 , 2 5  .
  • BThe function is negative at π‘₯ ∈ ο€Ό βˆ’ 2 5 , 2 5  , positive at π‘₯ ∈ ℝ βˆ’  βˆ’ 2 5 , 2 5  , and equals zero when π‘₯ ∈  βˆ’ 2 5 , 2 5  .
  • CThe function is positive for all π‘₯ ∈ ℝ βˆ’ { βˆ’ 2 } , and equals zero when π‘₯ = βˆ’ 2 .
  • DThe function is positive for all π‘₯ ∈ ℝ βˆ’  2 5  , and equals zero when π‘₯ = 2 5 .

Q22:

Determine the sign of the function 𝑓 ( π‘₯ ) = ( π‘₯ βˆ’ 8 ) ( π‘₯ βˆ’ 7 ) .

  • A The function is positive when π‘₯ ∈ ℝ βˆ’ [ 7 , 8 ] , the function is negative when π‘₯ ∈ ] 7 , 8 [ , and the function equals zero when π‘₯ ∈ { 7 , 8 } .
  • B The function is positive when π‘₯ ∈ ] 7 , 8 [ , the function is negative when π‘₯ ∈ ℝ βˆ’ [ 7 , 8 ] , and the function equals zero when π‘₯ ∈ { 7 , 8 } .
  • C The function is positive when π‘₯ ∈ [ 7 , 8 ] , the function is negative when π‘₯ ∈ ℝ βˆ’ { 7 , 8 } , and the function equals zero when π‘₯ ∈ { 7 , 8 } .
  • D The function is positive when π‘₯ ∈ ℝ βˆ’ { 7 , 8 } , the function is negative when π‘₯ ∈ [ 7 , 8 ] , and the function equals zero when π‘₯ ∈ { 7 , 8 } .

Q23:

Determine the interval in which the function 𝑓 ( π‘₯ ) = βˆ’ 1 5 βˆ’ 8 π‘₯ βˆ’ π‘₯ 2 is NOT negative.

  • A [ βˆ’ 5 , βˆ’ 3 ]
  • B ℝ
  • C [ 3 , 5 ]
  • D ℝ βˆ’ [ 3 , 5 ]
  • E ℝ βˆ’ [ βˆ’ 5 , βˆ’ 3 ]

Q24:

Determine the sign of the function 𝑓 ( π‘₯ ) = βˆ’ π‘₯ βˆ’ 2 π‘₯ βˆ’ 7 2 .

  • AThe function is negative for all π‘₯ ∈ ℝ .
  • BThe function is positive for all π‘₯ ∈ ℝ .
  • CThe function is negative for all π‘₯ ∈ ℝ βˆ’ { 0 } , and the function equals zero when π‘₯ = 0 .
  • DThe function is positive for all π‘₯ ∈ ℝ βˆ’ { 0 } , and the function equals zero when π‘₯ = 0 .

Q25:

Determine the sign of the function 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 1 6 π‘₯ + 6 4 2 .

  • AThe function is positive when π‘₯ ∈ ℝ βˆ’ { 8 } , and the function equals zero when π‘₯ = 8 .
  • BThe function is positive for all π‘₯ ∈ ℝ .
  • CThe function is positive when π‘₯ ∈ ℝ βˆ’ { βˆ’ 8 } , and the function equals zero when π‘₯ = βˆ’ 8 .
  • DThe function is positive when π‘₯ ∈ ℝ βˆ’ { βˆ’ 8 , 8 } , and the function equals zero when π‘₯ ∈ { βˆ’ 8 , 8 } .
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