U1 L1.2: Evaluating Functions

• Learning Objectives: At the end of this lesson, you should be able to evaluate functions
• Solution for Example 3

  To find 𝑇 1 and 𝑉 −2, evaluate them separately and then subtract the results. 𝑇 1 = 30, 𝑉 −2 = -8. Therefore, 𝑇 1 − 𝑉 −2 = 30 - (-8) = 38
• Evaluating Functions

  To evaluate a function means to replace (or substitute) the variable in the function with a given number or expression. In simplifying a series of operations, we use the GEMDAS Rule
• Solution for Example 1

  The expression 𝑓 2 means evaluating the function 𝑓 𝑥 = 2𝑥 + 7 using 𝑥 = 2. Replace 𝑥 in the function with 2: 𝑓 2 = 2(2) + 7 = 4 + 7 = 11
• Solution for Example 2

  The expression 𝑔 −3 means evaluating the function 𝑔 𝑥 = 𝑥0 + 4𝑥 − 5 using �� = −3. Replace �� with −3: 𝑔 −3 = −3(0) + 4(-3) - 5 = -8
• Topics to revisit before starting the lesson

  • Evaluating algebraic expressions
  • Operations on algebraic expressions
  • Order of Operations (GEMDAS)
• Therefore, 𝑻 𝟏 − 𝑽 −�� = 𝟑𝟔
• Evaluating 𝑇 1 − 𝑉 −2

  𝑇 1 − 𝑉 −2 = 30 − −6 = 36
• Example 4


  Evaluate 𝑓 2𝑥 + 1 if 𝑓 𝑥 = 3�� + 4
• Evaluating Functions

  In the previous examples, we evaluated functions by replacing the variable in the function by a number. In some cases, we evaluate a function by replacing the variable in the function by some expressions
• Evaluating 𝑇 1

  1. 𝑇 𝑥 = 3 𝑥 + 9

  2. 𝑇 1 = 3 1 + 9 = 3 10 = 30
• If 𝑻 𝒙 = 𝟐 �� − 𝟑 and �� �� = ��𝒙�� − 𝟓𝒙 + 𝟖, what is the value of 𝑻 𝟐 − 𝑽 −𝟏 ?
• Evaluating 𝑉 −2

  1. 𝑉 𝑥 = 2𝑥0 + 4𝑥 − 6

  2. 𝑉 −2 = 2 −2 0 + 4 −2 − 6 = 2 4 − 8 − 6 = 8 − 8 − 6 = −6
• Solution for Example 5

  To solve for 𝑔 𝑥 + ℎ, we need to replace the variable 𝑥 in the function 𝑔 𝑥 = 𝑥0 + 4𝑥 − 6 with the expression 𝑥 + ℎ. 𝑔 𝑥 + ℎ = 𝑥 + ℎ 0 + 4 𝑥 + ℎ − 6
• Evaluating Functions

  In order to find the value 𝑇 1 − 𝑉 −2, we need to evaluate �� 1 and �� −2 separately, and then subtract the results
• Example 5

  Given that 𝑔 𝑥 = 𝑥0 + 4𝑥 − 6, solve for 𝑔 𝑥 + ℎ
• Solution for Example 4

  The expression 𝑓 2𝑥 + 1 means that we are going to replace the variable 𝑥 in the function 𝑓 𝑥 = 3𝑥 + 4 with 2𝑥 + 1. �� 2�� + 1 = 3 2𝑥 + 1 + 4 = 6𝑥 + 3 + 4 = 6𝑥 + 7
• The Wealth of Nations was written

  1776
• Given that �� 𝒙 = 𝒙�� − ��𝒙 + 𝟗, solve for 𝒕 𝒙 + 𝒉
• To solve for 𝑔 𝑥 + ℎ, we need to replace the variable 𝑥 in the function 𝑔 𝑥 = 𝑥0 + 4𝑥 − 6 with the expression 𝑥 + ℎ
• Evaluating 𝑓 2

  1. Identify the location of 𝑥 = 2

  2. Locate where 𝑥 = 2 lies in the piecewise function

  3. Identify the value of 𝑓 2 by using the expression 5𝑥0 + 4
• Solving for 𝑔 𝑥 + ℎ

  1. Replace 𝑥 in 𝑔 𝑥 = 𝑥0 + 4𝑥 − 6 with 𝑥 + ℎ

  2. Simplify the algebraic expression to get the result 𝑔 𝑥 + ℎ = 𝑥0 + 2ℎ𝑥 + ℎ0 + 4𝑥 + 4ℎ − 6 = 𝑥𝟐 + 𝟐ℎ𝑥 + ℎ𝟐 + 𝟒𝑥 + 𝟒ℎ − 𝟔
• Evaluating 𝑓 0

  Identify the location of 𝑥 = 0 in the piecewise function
• In the next examples, we are going to learn how to evaluate piecewise functions
• Given the piecewise function 𝑓 𝑥 = E 𝑥F + 1, if 𝑥 ≤ 1, evaluate 𝑓 2 and 𝑓 0
• Try This!
• Given the piecewise function 𝑆 𝑥


  1. Evaluate the following:

  2. 𝑆 3

  3. 𝑆 −4

  4. 𝑆 6
• Evaluating Functions
• Solution
• Evaluating 𝑆(3)

  1. Locate 𝑥 = 3 in the piecewise function

  2. Use the expression 8 − 3�� − 4��0 to evaluate 𝑆 3 = 8 − 3 3 − 4 3 0 = 8 − 3 3 − 4 9 = 8 − 9 − 36 = −37
• Given the piecewise function 𝑆 𝑥

  1. Evaluate the following:

  2. �� −𝟑

  3. 𝑆 𝟐
• Evaluating 𝑆(6)

  Locate 𝑥 = 6 in the piecewise function
• Evaluating 𝑓 0

  1. Identify the location of 𝑥 = 0 in the piecewise function

  2. Identify the value of 𝑓 0 by using the expression 𝑥F + 1

  3. Calculate 𝑓 0 = 0 F + 1 = 0 + 1 = 1
• Given the piecewise function
• De La Salle Medical and Health Sciences Institute - Special Health Sciences Senior High School
• Evaluating 𝑆(−4)


  1. Locate 𝑥 = −4 in the piecewise function

  2. Use the expression −2 2�� − 4 to evaluate 𝑆 −4 = −2 2 −4 − 4 = −2 −8 − 4 = −2 −12 = 24
• Example 7:
• Evaluate the following:

  1. 𝑆 2

  2. 𝑆 -6

  3. 𝑆 9
• The expression 𝑓 1 means that we are looking for the 𝑦-value that corresponds to 𝑥 = 1
• Evaluating Functions


  We will learn how to evaluate functions based on their graphs