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Algebra 1

4.5.3 Comparing Functions

Algebra 14.5.3 Comparing Functions

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Activity

A college student is choosing between two data plans for her new cell phone. Both plans include an allowance of 2 gigabytes of data per month. The monthly cost of each option can be seen as a function and represented with an equation:

  • Option A: A ( x ) = 60 A ( x ) = 60
  • Option B:  B ( x ) = 10 x + 25 B ( x ) = 10 x + 25

In each function, the input, x x , represents the gigabytes of data used over the monthly allowance.

The student decides to find the cost of each option when she exceeds the 2 gigabytes of data allowed each month. Use the function rules to help her compare the data plans.

1. What is the value of A(1)

2. What is the value of B(1)

3. Explain which data plan seems to be the best option for when 1 additional gigabyte of data is used.

4. What is the value of A ( 7.5 ) A ( 7.5 ) ?

5. What is the value of B ( 7.5 ) B ( 7.5 ) ?

6. Explain which data plan seems to be the best option for when 7.5 additional gigabytes of data are used.

7. Describe option A, A ( x ) = 60 A ( x ) = 60 , in words.

8. Describe option B, B ( x ) = 10 x + 25 B ( x ) = 10 x + 25 , in words.

9. Use the desmos graphing tool or technology outside the course. Graph each function on the same coordinate plane.

10. Explain which plan you think she should choose.

11. The student only budgeted $50 a month for her cell phone. She thought, “I wonder how many gigabytes of data I would have for $50 if I go with Option B?” and wrote B ( x ) = 50 B ( x ) = 50 . How many additional gigabytes can the student use beyond the 2 allotted by the plan?

12. Explain how you knew how many gigabytes of data she would have for $50.

Are you ready for more?

Extending Your Thinking

Describe a different data plan that, for any amount of data used, would cost no more than one of the given plans and no less than the other given plan. Explain or show how you know this data plan would meet these requirements.

Self Check

Rental car company A charges $50 a day plus $0.50 a mile, m , and is modeled by function A .
  1. Company A is always more than Company B.
  2. A ( 100 ) < B ( 100 )
  3. A ( 100 ) = B ( 100 )
  4. A ( 100 ) > B ( 100 )

Additional Resources

Comparing Function Values

Functions are used in a variety of real-life situations. They are useful in making comparisons, especially when comparing the cost.

Example

The daily cost to printing company A to print a book is modeled by a function  C ( x ) = 3.25 x + 1500 C ( x ) = 3.25 x + 1500 , where  C C  is the total daily cost, in dollars, and x x  is the number of books printed.

Their competitor, company  B,  can print a book at a cost modeled by  C ( x ) = 2.75 x + 2500 C ( x ) = 2.75 x + 2500 .

Find C ( 1000 ) C ( 1000 ) for each company and make a statement about the result.

Company A: C ( 1000 ) = 3.25 ( 1000 ) + 1500 C ( 1000 ) = 3.25 ( 1000 ) + 1500

C = 4750 C = 4750

Company B: C ( 1000 ) = 2.75 ( 1000 ) + 2500 C ( 1000 ) = 2.75 ( 1000 ) + 2500

C = 5250 C = 5250

When 1,000 books are printed, Company A has the better price.

Try it

Try It: Comparing Function Values

The daily cost to manufacturing company A is modeled by the function C ( x ) = 7.25 x + 2500 C ( x ) = 7.25 x + 2500 , where C ( x ) C ( x ) is the total daily cost and x x  is the number of items manufactured.

The daily cost to manufacturing company B is modeled by the function C ( x ) = 8.5 x + 1500 C ( x ) = 8.5 x + 1500 .

Which company is less at C ( 1000 ) C ( 1000 ) ? Explain the meaning of the result.

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