Worksheet: Expressing Information in Terms of Functions

In this worksheet, we will practice expressing information in terms of functions.

Q1:

Find an expression for the area ๐ด of a square as a function of its diagonal length ๐‘‘ .

  • A ๐ด = ๐‘‘ โˆš 2
  • B ๐ด = ๐‘‘ 2 ๏Šจ
  • C ๐ด = ๐‘‘ ๏Šจ
  • D ๐ด = 2 ๐‘‘ ๏Šจ
  • E ๐ด = ๐‘‘ 2

Q2:

A raindrop hitting a lake makes a circular ripple. The radius, in inches, grows as a function of time, in minutes, according to 2 โˆš ๐‘ก + 1 . Find the area of the ripple as a function of time, and then determine the area of the ripple at 3 seconds.

  • A ๐ด ( ๐‘ก ) = 2 ๐œ‹ โˆš ๐‘ก + 1 inches, ๐ด ( 3 ) = 4 ๐œ‹ square inches
  • B ๐ด ( ๐‘ก ) = 4 ๐œ‹ ( ๐‘ก + 1 ) inches, ๐ด ( 3 ) = 1 6 ๐œ‹ square inches
  • C ๐ด ( ๐‘ก ) = 2 ๐œ‹ ( ๐‘ก + 1 ) inches, ๐ด ( 3 ) = 8 ๐œ‹ square inches
  • D ๐ด ( ๐‘ก ) = 4 ( ๐‘ก + 1 ) inches, ๐ด ( 3 ) = 1 6 square inches
  • E ๐ด ( ๐‘ก ) = 4 ๐œ‹ โˆš ๐‘ก + 1 inches, ๐ด ( 3 ) = 8 ๐œ‹ square inches

Q3:

A rectangular piece of land has a length ๐‘ฅ m and an area of 2,775 m2. Find a function to calculate the width and find the width when the length is 75 m.

  • A The function ๐‘“ ( ๐‘ฅ ) = 2 , 7 7 5 ๐‘ฅ and the width is 208,125 m.
  • BThe function ๐‘“ ( ๐‘ฅ ) = 2 , 7 7 5 ๐‘ฅ and the width is 37 m.
  • C The function ๐‘“ ( ๐‘ฅ ) = 2 , 7 7 5 โˆ’ ๐‘ฅ and the width is 2,700 m.
  • D The function ๐‘“ ( ๐‘ฅ ) = 2 , 7 7 5 2 ๐‘ฅ and the width is 18.5 m.

Q4:

The volume of a cylinder can be described by the function ๐‘‰ = ๐œ‹ ๐‘Ÿ โ„Ž ๏Šจ . Find a formula to describe the volume of a cylinder where the radius is three times the height.

  • A ๐‘‰ = ๐œ‹ ๐‘Ÿ 3 ๏Šฉ
  • B ๐‘‰ = 9 ๐œ‹ โ„Ž ๏Šจ
  • C ๐‘‰ = 3 ๐œ‹ โ„Ž ๏Šฉ
  • D ๐‘‰ = ๐œ‹ ๐‘Ÿ 9 ๏Šจ

Q5:

The radius of a circular oil slick is expanding at a rate of 20 meters per day. Express the area of the circle as a function of ๐‘‘ , the number of days elapsed.

  • A ๐ด ( ๐‘‘ ) = 4 0 ๐œ‹ ๐‘‘ ๏Šจ
  • B ๐ด ( ๐‘‘ ) = 4 0 0 ๐œ‹ ๐‘‘
  • C ๐ด ( ๐‘‘ ) = 4 0 0 ๐œ‹ ๐‘‘ ๏Šจ
  • D ๐ด ( ๐‘‘ ) = 2 0 ๐œ‹ ๐‘‘ ๏Šจ
  • E ๐ด ( ๐‘‘ ) = 2 0 0 ๐œ‹ ๐‘‘ ๏Šจ

Q6:

The number of honey jars, โ„Ž , produced by a swarm of bees, ๐‘ ,is given by โ„Ž = ๐‘“ ( ๐‘ ) . A particular forest has 3 swarms of bees that produced 12 jars of honey. Express this information in terms of the function ๐‘“ .

  • A ๐‘“ ( 3 ) = 1 2
  • B ๐‘“ ( 1 ) = 3
  • C ๐‘“ ( 1 2 ) = 3
  • D ๐‘“ ( โ„Ž ) = ๐‘

Q7:

An oil spill grows with time such that its boundary is always a circle. Suppose that the radius is given by ๐‘Ÿ ( ๐‘ก ) = 5 โˆ’ ๐‘ก as a function of time ๐‘ก . Express the area of the spill ๐ด ( ๐‘ก ) as a function of time.

  • A ๐ด ( ๐‘ก ) = 2 ๐œ‹ ( 5 โˆ’ ๐‘ก )
  • B ๐ด ( ๐‘ก ) = ( 5 โˆ’ ๐‘ก ) ๏Šจ
  • C ๐ด ( ๐‘ก ) = 2 ๐œ‹ ( ๐‘ก โˆ’ 5 )
  • D ๐ด ( ๐‘ก ) = ๐œ‹ ( 5 โˆ’ ๐‘ก ) ๏Šจ
  • E ๐ด ( ๐‘ก ) = ๐œ‹ ( ๐‘ก โˆ’ 5 ) ๏Šจ

Q8:

The number of cubic yards of dirt, ๐ท , needed to cover a garden whose area is ๐‘Ž square feet is given by ๐ท = ๐‘” ( ๐‘Ž ) . A garden with area 5 0 0 0 ft2 requires 50 yd3 of dirt. Express this information in terms of the function ๐‘” .

  • A 5 0 0 = ๐‘” ( 5 0 )
  • B 5 0 0 0 = ๐‘” ( 5 0 )
  • C 5 0 = ๐‘” ( 5 0 0 0 )
  • D 5 0 = ๐‘” ( 5 0 5 0 )
  • E 5 0 = ๐‘” ( 5 0 0 )

Q9:

A rectangle has a length of 10 units and a width of 8 units. Squares of 2 ๐‘ฅ by 2 ๐‘ฅ units are cut out of each corner, and then the sides are folded up to create an open box. Express the volume of the box as a polynomial function in terms of ๐‘ฅ .

  • A ๐‘‰ ( ๐‘ฅ ) = 4 ๐‘ฅ โˆ’ 3 6 ๐‘ฅ + 8 0 ๐‘ฅ ๏Šฉ ๏Šจ
  • B ๐‘‰ ( ๐‘ฅ ) = 4 ๐‘ฅ + 3 6 ๐‘ฅ + 8 0 ๐‘ฅ ๏Šฉ ๏Šจ
  • C ๐‘‰ ( ๐‘ฅ ) = 3 2 ๐‘ฅ โˆ’ 1 4 4 ๐‘ฅ + 1 6 0 ๐‘ฅ ๏Šฉ ๏Šจ
  • D ๐‘‰ ( ๐‘ฅ ) = 1 6 ๐‘ฅ + 7 2 ๐‘ฅ + 8 0 ๐‘ฅ ๏Šฉ ๏Šจ
  • E ๐‘‰ ( ๐‘ฅ ) = 1 6 ๐‘ฅ + 8 ๐‘ฅ + 8 0 ๐‘ฅ ๏Šฉ ๏Šจ

Q10:

A gardener has 200 feet of fencing which he can use to enclose an area for a rectangular garden. By putting the garden against one wall of the house, only three sides need to be fenced. Let ๐‘ฅ be the length of the side perpendicular to the wall of the house. Write a function in terms of ๐‘ฅ for the area of the resulting garden.

  • A ๐ด ( ๐‘ฅ ) = 2 ๐‘ฅ โˆ’ 2 0 0 ๐‘ฅ ๏Šจ
  • B ๐ด ( ๐‘ฅ ) = 2 0 0 ๐‘ฅ โˆ’ 2 ๐‘ฅ ๏Šจ
  • C ๐ด ( ๐‘ฅ ) = ๐‘ฅ ( 2 0 0 โˆ’ ๐‘ฅ )
  • D ๐ด ( ๐‘ฅ ) = 2 ๐‘ฅ ( 2 0 0 ๐‘ฅ )

Q11:

The amount of garbage ๐บ produced by a town with a population ๐‘ is given by ๐บ = ๐‘“ ( ๐‘ ) . ๐บ is measured in tons per week, and ๐‘ is measured in thousands of people. Suppose a town has a population of 40โ€‰000 and produces 13 tons of garbage each week. Express this information in terms of the function ๐‘“ .

  • A 5 3 = ๐‘“ ( 1 3 )
  • B 1 3 = ๐‘“ ( 4 0 0 0 0 )
  • C 4 0 = ๐‘“ ( 1 3 )
  • D 4 0 0 0 0 = ๐‘“ ( 1 3 )
  • E 1 3 = ๐‘“ ( 4 0 )

Q12:

Assigning test grades to students is an example of a function.

Which of the following is this function written in function notation?

  • A ๐บ (grade of student A on the test) = student A
  • B ๐บ (student A) = grade of student A on the test

What is the domain of the function?

  • AAll the possible grades
  • BStudents taking the test
  • CTest scores
  • DNonnegative numbers

What is the codomain of the function?

  • ANonnegative numbers
  • BTest scores
  • CStudents taking the test
  • DAll the possible grades

Q13:

If the input of the function ๐‘” is ๐‘ฅ , then the output of the function is .

  • A ๐‘ฅ ( ๐‘” )
  • B ๐‘”
  • C ๐‘” ( ๐‘ฅ )
  • D ๐‘ฅ

Q14:

Which of the following is equation ๐‘ฅ = 3 expressed as a function of ๐‘ฅ .

  • A ๐‘ฆ = 3
  • B ๐‘“ ( ๐‘ฅ ) = 3
  • C ๐‘“ ( ๐‘ฆ ) = 3
  • DThis cannot be expressed as a function of ๐‘ฅ .
  • E ๐‘ง = 3

Q15:

A rectangle has a length of 10 inches and a width of 6 inches. If the length is increased by ๐‘ฅ inches and the width is increased by twice that amount, express the area of the rectangle as a function of ๐‘ฅ .

  • A ๐ด ( ๐‘ฅ ) = 1 2 ๐‘ฅ + 1 2 0
  • B ๐ด ( ๐‘ฅ ) = 2 4 ๐‘ฅ + 1 2 0
  • C ๐ด ( ๐‘ฅ ) = 2 ๐‘ฅ + 1 6 ๐‘ฅ + 6 0 ๏Šจ
  • D ๐ด ( ๐‘ฅ ) = ๐‘ฅ + 1 1 ๐‘ฅ + 3 0 ๏Šจ
  • E ๐ด ( ๐‘ฅ ) = 2 ๐‘ฅ + 2 6 ๐‘ฅ + 6 0 ๏Šจ

Q16:

The volume ๐‘‰ of mercury in a particular thermometer is a function of the measured temperature ๐‘‡ . If the temperature is the input and the volume is the output of this function, does each unique temperature give rise to a specific volume?

  • Ano
  • Byes

Q17:

A square has sides of length 12. Squares measuring ๐‘ฅ + 1 by ๐‘ฅ + 1 are cut out of each corner, and then the sides are folded up to create an open box. Express the volume of the box as a function in terms of ๐‘ฅ .

  • A ๐‘‰ ( ๐‘ฅ ) = 4 ๐‘ฅ + 4 4 ๐‘ฅ + 1 2 1 ๐‘ฅ ๏Šฉ ๏Šจ
  • B ๐‘‰ ( ๐‘ฅ ) = 4 ๐‘ฅ + 4 0 ๐‘ฅ + 1 0 0 ๐‘ฅ ๏Šฉ ๏Šจ
  • C ๐‘‰ ( ๐‘ฅ ) = 4 ๐‘ฅ โˆ’ 3 6 ๐‘ฅ + 6 0 ๐‘ฅ + 1 0 0 ๏Šฉ ๏Šจ
  • D ๐‘‰ ( ๐‘ฅ ) = 4 ๐‘ฅ โˆ’ 2 0 ๐‘ฅ + 1 0 0 ๐‘ฅ + 6 0 ๏Šฉ ๏Šจ
  • E ๐‘‰ ( ๐‘ฅ ) = 4 ๐‘ฅ โˆ’ 4 4 ๐‘ฅ + 1 2 1 ๐‘ฅ + 1 0 0 ๏Šฉ ๏Šจ

Q18:

Given that ๐ด is the area of a circle, and ๐‘Ÿ is its radius, express ๐ด as a function of ๐‘Ÿ , and determine the value of ๐ด ( 1 2 ) giving your answer in terms of ๐œ‹ if necessary.

  • A ๐ด ( ๐‘Ÿ ) = 2 ๐œ‹ ๐‘Ÿ , ๐ด ( 1 2 ) = 2 4 ๐œ‹
  • B ๐ด ( ๐‘Ÿ ) = ๐‘Ÿ , ๐ด ( 1 2 ) = 1 2
  • C ๐ด ( ๐‘Ÿ ) = ๐‘Ÿ ๏Šจ , ๐ด ( 1 2 ) = 1 4 4
  • D ๐ด ( ๐‘Ÿ ) = ๐œ‹ ๐‘Ÿ ๏Šจ , ๐ด ( 1 2 ) = 1 4 4 ๐œ‹
  • E ๐ด ( ๐‘Ÿ ) = 1 2 ๐‘Ÿ ๏Šจ , ๐ด ( 1 2 ) = 1 , 7 2 8

Q19:

A rectangle is twice as long as it is wide. Squares of length 2 units are cut out from each corner. Then, the sides are folded up to make an open box. Express the volume of the box as a function of the width ( ๐‘ฅ ) .

  • A ๐‘‰ ( ๐‘ฅ ) = 4 ๐‘ฅ + 2 4 ๐‘ฅ + 3 2 ๏Šจ
  • B ๐‘‰ ( ๐‘ฅ ) = 2 ๐‘ฅ + 4 ๐‘ฅ + 1 6 ๏Šจ
  • C ๐‘‰ ( ๐‘ฅ ) = 4 ๐‘ฅ + 8 ๐‘ฅ + 1 6 ๏Šจ
  • D ๐‘‰ ( ๐‘ฅ ) = 2 ๐‘ฅ โˆ’ 1 2 ๐‘ฅ + 1 6 ๏Šจ
  • E ๐‘‰ ( ๐‘ฅ ) = 4 ๐‘ฅ โˆ’ 2 4 ๐‘ฅ + 3 2 ๏Šจ

Q20:

A rectangle has a perimeter of 36. Find a function ๐ด ( ๐‘ค ) to describe the area of the rectangle, based upon its width.

  • A ๐ด ( ๐‘ค ) = 3 6 ๐‘ค ๏Šจ
  • B ๐ด ( ๐‘ค ) = ๐‘ค ( 3 6 โˆ’ 2 ๐‘ค )
  • C ๐ด ( ๐‘ค ) = 1 8 ๐‘ค โˆ’ ๐‘ค ๏Šจ
  • D ๐ด ( ๐‘ค ) = 8 1 ๐‘ค

Q21:

A cube is increasing in size. Initially, an edge measured 3 feet, and it increases at a rate of 2 feet per minute. Find an expression for the volume of the cube, ๐‘‰ ( ๐‘š ) , as a function of the number of minutes elapsed, ๐‘š . Write your answer as a polynomial in standard form.

  • A ๐‘‰ ( ๐‘š ) = ๐‘š + 3 6 ๐‘š + 5 4 ๐‘š + 2 7 ๏Šฉ ๏Šจ
  • B ๐‘‰ ( ๐‘š ) = 8 ๐‘š + 2 4 ๐‘š + 5 4 ๐‘š + 1 8 ๏Šฉ ๏Šจ
  • C ๐‘‰ ( ๐‘š ) = 8 ๐‘š + 3 6 ๐‘š + 5 4 ๐‘š + 2 7 ๏Šฉ ๏Šจ
  • D ๐‘‰ ( ๐‘š ) = 9 ๐‘š + 2 4 ๐‘š + 5 4 ๐‘š + 2 7 ๏Šฉ ๏Šจ
  • E ๐‘‰ ( ๐‘š ) = 8 ๐‘š + 3 6 ๐‘š + 2 7 ๐‘š + 2 7 ๏Šฉ ๏Šจ

Q22:

Rewrite the following in terms of a function ๐‘“ , using the language of inputs and outputs.

The output is greater than 5 when the input is 7.

  • A ๐‘“ ( 5 ) > 7
  • B ๐‘“ ( 7 ) = 5
  • C ๐‘“ ( 7 ) < 5
  • D ๐‘“ ( 5 ) < 7
  • E ๐‘“ ( 7 ) > 5

The output at input ๐‘Ž + ๐‘ is the same as the sum of the outputs at ๐‘Ž and ๐‘ .

  • A ๐‘“ ( ๐‘Ž ) = ๐‘“ ( ๐‘Ž + ๐‘ )
  • B ๐‘“ ( ๐‘Ž + ๐‘ ) = ๐‘“ ( ๐‘ ) โˆ’ ๐‘“ ( ๐‘Ž )
  • C ๐‘“ ( ๐‘Ž + ๐‘ ) = ๐‘“ ( ๐‘Ž ) + ๐‘“ ( ๐‘ )
  • D ๐‘“ ( ๐‘ ) = ๐‘“ ( ๐‘Ž + ๐‘ )
  • E ๐‘“ ( ๐‘Ž + ๐‘ ) = ๐‘“ ( ๐‘Ž ) โˆ’ ๐‘“ ( ๐‘ )

The outputs for inputs ๐‘ and ๐‘‘ are the same.

  • A ๐‘“ ( ๐‘ ) = ๐‘“ ( ๐‘‘ )
  • B ๐‘“ ( ๐‘ ) ร— ๐‘“ ( ๐‘‘ ) = 1
  • C ๐‘“ ( ๐‘ ) รท ๐‘“ ( ๐‘‘ ) = ๐‘“ ( ๐‘ )
  • D ๐‘“ ( ๐‘ ) + ๐‘“ ( ๐‘‘ ) = 1
  • E ๐‘“ ( ๐‘ ) โˆ’ ๐‘“ ( ๐‘‘ ) = 1

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