Functional Notation and Terminology: Solutions

Here are some solutions to the Exercises to accompany the section Functional Notation and Terminology.  

Functional Evaluation

  1. Evaluate each of the following functions at the given input value:
    1. Evaluate the function given by {(1, 0), (0, 0), (2, -0.4), (-1, -0.4), (3, -1.2), (-2, -1.2), (-3, -2.4)} using an input of 3.
      Solution
      The only ordered pair with a first coordinate of 3 is (3, -1.2), which has a second coordinate of -1.2, which is the corresponding "output".
       
    2. Evaluate the following function at t = 0.8:
      t h
      0
      0.2
      0.4
      0.6
      0.8
      1.
      1.2
      1.4      		 1.4
      2.4
      3
      3.2
      3
      2.4
      1.4
      0
      Solution
      The value t = 0.8 appears in the fifth row.  The corresponding entry in the "output" column is 3.
       
    3. Evaluate the following function using an input of b:

      4. Solution
      Following the arrow from b, on the left, we reach the letter g, on the right.
       
    4. Evaluate the following function using an input of 2:

      Note: You will only be able to give an approximate answer.

      Solution
      Finding the value 2 on the horizontal axis, traveling directly upward to the graph, and then directly to the left until we hit the vertical axis:

       

      we are left at the value 12 on the vertical axis (as indicated by the tick marks on the axis; five marks between 10 and 12, means that each are 2 units apart; since we are one mark up from 10, we must be at 10 + 2 = 12), which is desired the output value.

    5. Evaluate the function given by the equation, h = 1.4 + 6t - 5t2, at t = 0.2.
      Solution
      Plugging in t = 0.2 gives h = 1.4 + 6(0.2) - 5(0.2)2 = 1.4 + 1.2 - 5(.04) = 2.6 - 0.2 = 2.4 as our output.  Note: When simplifying, make sure to follow the correct order of operations.
    6. Evaluate the following function using an input of x = 2:

      Solution
      As before, we plug in x = 2 to get (2 + 5)(2) = (7)(2) = 14 as our output.  We could represent this pictorially as:

    7. Evaluate the function given by the rule: "Add 5 to the input, then multiply the result by the input," using an input of 3.
      Solution
      This is actually the same rule as in part f).  Literally, we would take our input of 3, add 5, to get 8, then multiply that 8 by 3 to get 24 as our output.  

    Back to Exercises.

Functional Terminology

  1. For each of following functions, describe the set of values in the domain and range:
    1. The function with the arrow diagram:

      2.  
      Solution
      The domain is {a, b, c, d}, while the range is {e, g, h}. 
    2. The function given by the following table:
      x y
      -3
      -2
      -1
      0
      1
      2       		5
      1
      -2
      1
      3
      4
      Solution
      The domain is {-3, -2, -1, 0, 1, 2}, while the range is {-2, 1, 3, 4, 5}.

    3. The function given by the following plot:

      Solution
      If we trace along the graph:
      We can see that the domain is all values between -2 and 2, while the range is is all values between 1 and 4.  We can write this in set notation as domain = {x | -2 ≤ x ≤ 2} and range =  {y | 1 ≤ y ≤ 4}.

    Back to Exercises.

Functional Notation

  1. For each of the following functions, simplify the given functional expression:
    1. If the function f is given by {(1, 0), (0, 0), (2, -0.4), (-1, -0.4), (3, -1.2), (-2, -1.2), (-3, -2.4)}, compute f(2).
      Solution
      The only ordered pair with a first coordinate of 2 is (2, -0.4), which has a second coordinate of -0.4, so f(2) = -0.4.
       
    2. If the function g is given by the table
      t u = g(t)
      0
      0.2
      0.4
      0.6
      0.8
      1.
      1.2
      1.4      		 1.4
      2.4
      3
      3.2
      3
      2.4
      1.4
      0
      compute g(0.4).
      Solution
      The value t = 0.4 appears in the third row.  The corresponding entry in the "output" column is u = 3, so g(0.4) = 3.
       
    3. If the function h is given by the arrow diagram:

      4. compute h(d).
      Solution
      Following the arrow from d, on the left, we reach the letter g, on the right, so h(d) = g.  Note: The letter "h" in the arrow diagram should not be confused with the name of the function itself; this is another time when it is important to pay attention to the context in which a letter appears.
    4. If the function k is given by the graph:

      compute k(-2).  Note: You will only be able to give an approximate answer.

      Solution
      Finding the value -2 on the horizontal axis, traveling directly upward to the graph, and then directly to the left until we hit the vertical axis:

       

      we are left at the value 12 on the vertical axis, so k(-2) = 12.  

    5. If the function s is given by the equation, s(t) = 1.4 + 6t - 5t2, compute s(0.4).
      Solution
      Plugging in t = 0.4 gives h = s(0.4) = 1.4 + 6(0.4) - 5(0.4)2 = 1.4 + 2.4 - 5(.16) = 3.8 - 0.8 = 3 as our output, so s(0.4) = 3.
    6. If the function q may be pictured by the following black-box diagram:

      compute q(3).

      Solution
      Plugging in x = 3 gives (3 + 5)(3) = (8)(3) = 24 as our output, so q(3) = 24.
    7. If the function r is given by the rule: "Add 5 to the input, then multiply the result by the input," compute r(2).
      Solution
      Taking our input of 2, adding 5, gives 7; multiplying that by 2 gives 14, so r(2) = 14.
    8. Compute id(5), where id is the identity function.
      Solution
      Since, by definition, this function simply returns the input, it will return the value 5, that is, id(5) = 5.

    Back to Exercises.

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