Worksheet: Forming Functions

In this worksheet, we will practice forming and deriving the equation of a function from different situations.


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

  • A๐ด=๐‘‘2
  • B๐ด=๐‘‘๏Šจ
  • C๐ด=2๐‘‘๏Šจ
  • D๐ด=๐‘‘โˆš2
  • E๐ด=๐‘‘2๏Šจ


A raindrop hitting a lake makes a circular ripple. The radius, in inches, grows as a function of time, in minutes, according to 2โˆš๐‘ก+1. Find the area of the ripple as a function of time, and then determine the area of the ripple at 3 seconds.

  • A๐ด(๐‘ก)=4(๐‘ก+1) inches, ๐ด(3)=16 square inches
  • B๐ด(๐‘ก)=2๐œ‹โˆš๐‘ก+1 inches, ๐ด(3)=4๐œ‹ square inches
  • C๐ด(๐‘ก)=4๐œ‹(๐‘ก+1) inches, ๐ด(3)=16๐œ‹ square inches
  • D๐ด(๐‘ก)=2๐œ‹(๐‘ก+1) inches, ๐ด(3)=8๐œ‹ square inches
  • E๐ด(๐‘ก)=4๐œ‹โˆš๐‘ก+1 inches, ๐ด(3)=8๐œ‹ square inches


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.

  • AThe function ๐‘“(๐‘ฅ)=2,7752๐‘ฅ and the width is 18.5 m.
  • BThe function ๐‘“(๐‘ฅ)=2,775๐‘ฅ and the width is 208,125 m.
  • CThe function ๐‘“(๐‘ฅ)=2,775โˆ’๐‘ฅ and the width is 2,700 m.
  • DThe function ๐‘“(๐‘ฅ)=2,775๐‘ฅ and the width is 37 m.


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

  • A๐‘‰=๐œ‹๐‘Ÿ9๏Šจ
  • B๐‘‰=3๐œ‹โ„Ž๏Šฉ
  • C๐‘‰=๐œ‹๐‘Ÿ3๏Šฉ
  • D๐‘‰=9๐œ‹โ„Ž๏Šจ


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.

  • A๐ด(๐‘‘)=400๐œ‹๐‘‘
  • B๐ด(๐‘‘)=400๐œ‹๐‘‘๏Šจ
  • C๐ด(๐‘‘)=200๐œ‹๐‘‘๏Šจ
  • D๐ด(๐‘‘)=40๐œ‹๐‘‘๏Šจ
  • E๐ด(๐‘‘)=20๐œ‹๐‘‘๏Šจ


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‘“.

  • A๐‘“(12)=3
  • B๐‘“(1)=3
  • C๐‘“(โ„Ž)=๐‘
  • D๐‘“(3)=12


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)=5โˆ’๐‘ก as a function of time ๐‘ก. Express the area of the spill ๐ด(๐‘ก) as a function of time.

  • A๐ด(๐‘ก)=2๐œ‹(5โˆ’๐‘ก)
  • B๐ด(๐‘ก)=(5โˆ’๐‘ก)๏Šจ
  • C๐ด(๐‘ก)=๐œ‹(5โˆ’๐‘ก)๏Šจ
  • D๐ด(๐‘ก)=2๐œ‹(๐‘กโˆ’5)
  • E๐ด(๐‘ก)=๐œ‹(๐‘กโˆ’5)๏Šจ


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 5,000 ft2 requires 50 yd3 of dirt. Express this information in terms of the function ๐‘”.

  • A50=๐‘”(5,050)
  • B500=๐‘”(50)
  • C50=๐‘”(5,000)
  • D5,000=๐‘”(50)
  • E50=๐‘”(500)


A rectangle has a length of 10 units and a width of 8 units. Squares of 2๐‘ฅ by 2๐‘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.

  • A๐‘‰(๐‘ฅ)=4๐‘ฅ+36๐‘ฅ+80๐‘ฅ๏Šฉ๏Šจ
  • B๐‘‰(๐‘ฅ)=16๐‘ฅ+72๐‘ฅ+80๐‘ฅ๏Šฉ๏Šจ
  • C๐‘‰(๐‘ฅ)=4๐‘ฅโˆ’36๐‘ฅ+80๐‘ฅ๏Šฉ๏Šจ
  • D๐‘‰(๐‘ฅ)=16๐‘ฅ+8๐‘ฅ+80๐‘ฅ๏Šฉ๏Šจ
  • E๐‘‰(๐‘ฅ)=32๐‘ฅโˆ’144๐‘ฅ+160๐‘ฅ๏Šฉ๏Šจ


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.

  • A๐ด(๐‘ฅ)=๐‘ฅ(200โˆ’๐‘ฅ)
  • B๐ด(๐‘ฅ)=2๐‘ฅ(200๐‘ฅ)
  • C๐ด(๐‘ฅ)=200๐‘ฅโˆ’2๐‘ฅ๏Šจ
  • D๐ด(๐‘ฅ)=2๐‘ฅโˆ’200๐‘ฅ๏Šจ


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 40,000 and produces 13 tons of garbage each week. Express this information in terms of the function ๐‘“.

  • A40,000=๐‘“(13)
  • B40=๐‘“(13)
  • C53=๐‘“(13)
  • D13=๐‘“(40)
  • E13=๐‘“(40,000)


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

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

  • A๐บ(grade of student A on the test) = student A
  • B๐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?

  • ATest scores
  • BAll the possible grades
  • CStudents taking the test
  • DNonnegative numbers


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

  • A๐‘ฅ(๐‘”)
  • B๐‘ฅ
  • C๐‘”
  • D๐‘”(๐‘ฅ)


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

  • A๐‘“(๐‘ฅ)=3
  • B๐‘ง=3
  • CThis cannot be expressed as a function of ๐‘ฅ.
  • D๐‘“(๐‘ฆ)=3
  • E๐‘ฆ=3


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.

  • A๐ด(๐‘ฅ)=2๐‘ฅ+26๐‘ฅ+60๏Šจ
  • B๐ด(๐‘ฅ)=24๐‘ฅ+120
  • C๐ด(๐‘ฅ)=๐‘ฅ+11๐‘ฅ+30๏Šจ
  • D๐ด(๐‘ฅ)=12๐‘ฅ+120
  • E๐ด(๐‘ฅ)=2๐‘ฅ+16๐‘ฅ+60๏Šจ


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


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.

  • A๐‘‰(๐‘ฅ)=4๐‘ฅโˆ’44๐‘ฅ+121๐‘ฅ+100๏Šฉ๏Šจ
  • B๐‘‰(๐‘ฅ)=4๐‘ฅโˆ’20๐‘ฅ+100๐‘ฅ+60๏Šฉ๏Šจ
  • C๐‘‰(๐‘ฅ)=4๐‘ฅ+44๐‘ฅ+121๐‘ฅ๏Šฉ๏Šจ
  • D๐‘‰(๐‘ฅ)=4๐‘ฅ+40๐‘ฅ+100๐‘ฅ๏Šฉ๏Šจ
  • E๐‘‰(๐‘ฅ)=4๐‘ฅโˆ’36๐‘ฅ+60๐‘ฅ+100๏Šฉ๏Šจ


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(12) giving your answer in terms of ๐œ‹ if necessary.

  • A๐ด(๐‘Ÿ)=๐‘Ÿ, ๐ด(12)=12
  • B๐ด(๐‘Ÿ)=12๐‘Ÿ๏Šจ, ๐ด(12)=1,728
  • C๐ด(๐‘Ÿ)=๐œ‹๐‘Ÿ๏Šจ, ๐ด(12)=144๐œ‹
  • D๐ด(๐‘Ÿ)=2๐œ‹๐‘Ÿ, ๐ด(12)=24๐œ‹
  • E๐ด(๐‘Ÿ)=๐‘Ÿ๏Šจ, ๐ด(12)=144


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).

  • A๐‘‰(๐‘ฅ)=2๐‘ฅโˆ’12๐‘ฅ+16๏Šจ
  • B๐‘‰(๐‘ฅ)=4๐‘ฅ+24๐‘ฅ+32๏Šจ
  • C๐‘‰(๐‘ฅ)=4๐‘ฅโˆ’24๐‘ฅ+32๏Šจ
  • D๐‘‰(๐‘ฅ)=2๐‘ฅ+4๐‘ฅ+16๏Šจ
  • E๐‘‰(๐‘ฅ)=4๐‘ฅ+8๐‘ฅ+16๏Šจ


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.

  • A๐ด(๐‘ค)=81๐‘ค
  • B๐ด(๐‘ค)=36๐‘ค๏Šจ
  • C๐ด(๐‘ค)=18๐‘คโˆ’๐‘ค๏Šจ
  • D๐ด(๐‘ค)=๐‘ค(36โˆ’2๐‘ค)


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.

  • A๐‘‰(๐‘š)=8๐‘š+36๐‘š+54๐‘š+27๏Šฉ๏Šจ
  • B๐‘‰(๐‘š)=8๐‘š+36๐‘š+27๐‘š+27๏Šฉ๏Šจ
  • C๐‘‰(๐‘š)=9๐‘š+24๐‘š+54๐‘š+27๏Šฉ๏Šจ
  • D๐‘‰(๐‘š)=๐‘š+36๐‘š+54๐‘š+27๏Šฉ๏Šจ
  • E๐‘‰(๐‘š)=8๐‘š+24๐‘š+54๐‘š+18๏Šฉ๏Šจ


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.

  • A๐‘“(5)<7
  • B๐‘“(5)>7
  • C๐‘“(7)<5
  • D๐‘“(7)=5
  • E๐‘“(7)>5

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

  • A๐‘“(๐‘Ž+๐‘)=๐‘“(๐‘Ž)+๐‘“(๐‘)
  • B๐‘“(๐‘Ž+๐‘)=๐‘“(๐‘Ž)โˆ’๐‘“(๐‘)
  • C๐‘“(๐‘)=๐‘“(๐‘Ž+๐‘)
  • D๐‘“(๐‘Ž)=๐‘“(๐‘Ž+๐‘)
  • E๐‘“(๐‘Ž+๐‘)=๐‘“(๐‘)โˆ’๐‘“(๐‘Ž)

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

  • A๐‘“(๐‘)รท๐‘“(๐‘‘)=๐‘“(๐‘)
  • B๐‘“(๐‘)ร—๐‘“(๐‘‘)=1
  • C๐‘“(๐‘)โˆ’๐‘“(๐‘‘)=1
  • D๐‘“(๐‘)+๐‘“(๐‘‘)=1
  • E๐‘“(๐‘)=๐‘“(๐‘‘)


A right circular cone has a radius of 3๐‘ฅ+6 and its height is 3 units less than its radius. Express the volume of the cone as a polynomial function, knowing that the volume of a cone with radius ๐‘Ÿ and height โ„Ž is ๐‘‰=13๐œ‹๐‘Ÿโ„Ž๏Šจ.

  • A๐‘‰(๐‘ฅ)=๐œ‹๏€น9๐‘ฅ+36๐‘ฅ+72๐‘ฅ+36๏…๏Šฉ๏Šจ
  • B๐‘‰(๐‘ฅ)=๐œ‹๏€น9๐‘ฅ+45๐‘ฅ+72๐‘ฅ+36๏…๏Šฉ๏Šจ
  • C๐‘‰(๐‘ฅ)=๐œ‹๏€น3๐‘ฅ+9๐‘ฅ+6๏…๏Šจ
  • D๐‘‰(๐‘ฅ)=๐œ‹๏€น9๐‘ฅ+45๐‘ฅ+36๐‘ฅ+36๏…๏Šฉ๏Šจ
  • E๐‘‰(๐‘ฅ)=๐œ‹๏€น๐‘ฅ+3๐‘ฅ+2๏…๏Šจ


A portion of fencing 100 feet long is cut into two pieces. One piece, which is ๐‘ฅ feet long, is used to enclose a square pen. The other piece is shaped into an enclosure as an equilateral triangle. What is the total area enclosed as a function of ๐‘ฅ?

  • A๐ด(๐‘ฅ)=๏€ป๐‘ฅ4๏‡+12๏€ผ100โˆ’๐‘ฅ3๏ˆ๏€ผ100โˆ’๐‘ฅ3๏ˆ๏Šจ
  • B๐ด(๐‘ฅ)=๏€พ๐‘ฅ4๏Š+(100โˆ’๐‘ฅ)6โˆš3๏Šจ๏Šจ
  • C๐ด(๐‘ฅ)=๐‘ฅ16+(100โˆ’๐‘ฅ)โˆš336๏Šจ๏Šจ
  • D๐ด(๐‘ฅ)=๐‘ฅ+(100โˆ’4๐‘ฅ)โˆš3๏Šจ


Write an equation that describes the relationship between the input and output.

Input (๐‘ฅ)026
Output (๐‘ฆ)0618
  • A๐‘ฆ=3๐‘ฅ+3
  • B๐‘ฆ=๐‘ฅโˆ’3
  • C๐‘ฆ=3๐‘ฅ
  • D๐‘ฆ=5๐‘ฅ
  • E๐‘ฆ=๐‘ฅ+3

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