Math Problem Statement

Suppose a function f(x) is defined by the given coordinates in the table below. Determine the values A, B, C, and D in the table for the function g(x) = 2f(x) + 5.

Solution

The problem asks to determine the values of AA, BB, CC, and DD for the function g(x)=2f(x)+5g(x) = 2f(x) + 5, based on the provided table for f(x)f(x).

The values of f(x)f(x) at different points x=2,0,1,5x = -2, 0, 1, 5 are given, and we are asked to compute the corresponding values of g(x)g(x) using the equation g(x)=2f(x)+5g(x) = 2f(x) + 5.

Step-by-step Calculation:

  1. For x=2x = -2:

    • f(2)=4f(-2) = -4
    • g(2)=2f(2)+5=2(4)+5=8+5=3g(-2) = 2f(-2) + 5 = 2(-4) + 5 = -8 + 5 = -3
    • Therefore, A=3A = -3.
  2. For x=0x = 0:

    • f(0)=1f(0) = -1
    • g(0)=2f(0)+5=2(1)+5=2+5=3g(0) = 2f(0) + 5 = 2(-1) + 5 = -2 + 5 = 3
    • Therefore, B=3B = 3.
  3. For x=1x = 1:

    • f(1)=3f(1) = 3
    • g(1)=2f(1)+5=2(3)+5=6+5=11g(1) = 2f(1) + 5 = 2(3) + 5 = 6 + 5 = 11
    • Therefore, C=11C = 11.
  4. For x=5x = 5:

    • f(5)=7f(5) = 7
    • g(5)=2f(5)+5=2(7)+5=14+5=19g(5) = 2f(5) + 5 = 2(7) + 5 = 14 + 5 = 19
    • Therefore, D=19D = 19.

Final Answers:

  • A=3A = -3
  • B=3B = 3
  • C=11C = 11
  • D=19D = 19

Would you like a more detailed explanation on any step? Here are some related questions for further practice:

  1. If g(x)=3f(x)+4g(x) = 3f(x) + 4, what would AA, BB, CC, and DD be?
  2. What happens to g(x)g(x) if f(x)f(x) is multiplied by a different constant?
  3. Can you describe how the graph of g(x)g(x) relates to the graph of f(x)f(x)?
  4. How would you solve if the function g(x)g(x) was given as g(x)=f(x)2g(x) = f(x) - 2?
  5. If g(x)g(x) was a quadratic function, what would the new values of AA, BB, CC, and DD be?

Tip: When modifying functions, always check if a transformation affects the slope, the intercept, or both.

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Math Problem Analysis

Mathematical Concepts

Linear Transformations
Functions

Formulas

g(x) = 2f(x) + 5

Theorems

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Suitable Grade Level

Grades 9-10