U1 L1.4: Solving Problems Involving Functions

• Topics to revisit before starting the lesson

  • Represent Real-Life Situations Using Functions
  • Evaluation Functions
  • Operations on Functions
• Solution for Example 1

  The function that describes the total cost of buying 𝑥 hotdog sandwiches is 𝐶 𝑥 = 20𝑥. Evaluating 𝐶 8 gives a cost of ₱160
• Learning Objectives: At the end of this lesson, you should be able to solve problems involving functions
• Example 1

  • A hotdog store on a street sells a hotdog sandwich that costs ₱20. Represent the cost 𝐶 of buying 𝑥 number of hotdog sandwich. Using this function, solve for the cost of buying 8 hotdog sandwiches.
• Example 2

  • A barangay captain wanted to enclose a rectangular piece of land next to a river using 100 meters of fencing materials. Represent the area 𝐴 of the rectangular piece of land in terms of its length, 𝑥. Solve for the area given that the length is 45 meters.
• Solution for Example 2

  The perimeter of the land is 𝑥 + 2𝑦. Since the perimeter is 100 meters, the equation is 𝑥 + 2�� = 100. The area of the land can be calculated based on this equation
• Try This


  • An accessories store on a mall sells a necklace that costs ₱300. Represent the cost 𝑪 of buying 𝒙 number of necklace. Using this function, solve for the cost of buying 6 pieces of necklace.
• A barangay captain wanted to enclose a rectangular piece of land next to a river to avoid the residents from getting to the river because it is too deep. He intends to use 200 meters of fencing materials to enclose the land.
• Solving for the volume of the box if the width measures 5 inches

  𝑉 = 4(5)^2 - 8(5) = 100 - 40 = 60 cubic inches
• Representing the area 𝑨 of the rectangular piece of land in terms of its width, 𝒚

  𝐴 = ��𝑦
• The perimeter of the land is 𝑥 + 2𝑦 = 100
• Volume of a rectangular box

  𝑉 = length × width × height
• Replacing 𝑦 in terms of 𝑥

  𝑦 = 100 − 𝑥 / 2
• Constructing a function 𝑉 that represents the volume of the box

  𝑉 = 4��(𝑥 − 2) = 4𝑥^2 - 8𝑥
• Solving for the area of the land when the length is 45 meters

  𝐴 45 = 1,237.5 m^2
• Area of a rectangular land

  𝐴 = ��𝑦
• The length of a rectangular box is five times its width while its height is 2 inches more than its width
• A van can be rented for ₱2500 for 12 hours or less. An additional charge of ₱100 per hour is charged if the van will be rented for more than 12 hours
• Volume of the box if the width measures 4 inches is 300 cubic inches
• Construct a function V that will represent the volume of the box
• Construct a function V that represents the cost of renting the van in terms of the number of hours t
• To solve for the volume of the box given that the width is 5 inches, evaluate V(5)
• Volume of the box is 300 cubic inches
• Volume of the box is V = 4x^2(x-2)
• If t > 12, the cost will be ₱2500 plus ₱100 per hour in excess of 12 hours. The excess hour can be expressed as t - 12. Thus, the second expression is 2500 + 100(t - 12)
• How much will be charged if the van is rented for 18 hours is ₱2700
• The desired piecewise function for V is V(t) = 2500 if 0 < t ≤ 12, 2500 + 100(t - 12) if t > 12
• If 0 < t ≤ 12, the cost will be ₱2500. Thus, the first expression of the piecewise function is simply 2500 (a constant)
• Volume of a rectangular prism is the product of its length, width, and height
• Conditions for the piecewise function V
• Volume of the box

  V = lwh
• Construct a function 𝑇 that represents the cost of renting the van in terms of the number of hours 𝑡

  𝑇 𝑡 = ₱3 000 if 0 < 𝑡 ≤ 12 and 𝑇 𝑡 = ₱3 000 + ₱150(�� - 12) if 𝑡 > 12
• When the van is rented for 18 hours, the total cost is ₱3 100
• The desired piecewise function will be 𝑉 𝑡 = 2 500 if 0 < 𝑡 ≤ 12 and 𝑉 𝑡 = 2 500 + 100 �� − 12 if 𝑡 > 12
• A van can be rented for ₱3 000 for 12 hours or less. An additional charge of ₱150 per hour is charged if the van will be rented for more than 12 hours
• Construct a function �� that represents the cost of a tricycle ride in terms of the number of kilometers, 𝑘


  𝑇 𝑘 = ₱20 if 0 < 𝑘 ≤ 3 and 𝑇 𝑘 = ₱20 + ₱3(𝑘 - 3) if 𝑘 > 3
• A tricycle ride costs ₱20 for the first 3 kilometers. An amount of ₱3 is charged for every kilometer in excess of 3 kilometers
• Excess hour

  𝑡 − 12 (actual time 𝑡 minus 12 hours is the excess hour)
• For 𝑡 > 12, the cost will be ₱2 500 plus ₱100 per hour in excess of 12 hours
• Cost for grooming a 50-lb dog is to be determined