Math Problem Statement

We are given two functions f(x) and g(x), which are related by the equation: g(x) = f(x+1) - 2. The table provides values of f(x) for x = 1, 2, 3, 4, 5, 6, and we need to compute the corresponding values of g(x) for x = 1, 2, 3, 4, 5.

Solution

Let us analyze and solve this question.

Problem Analysis:

We are given two functions f(x)f(x) and g(x)g(x), which are related by the equation: g(x)=f(x+1)−2g(x) = f(x+1) - 2

The table provides values of f(x)f(x) for x=1,2,3,4,5,6x = 1, 2, 3, 4, 5, 6, and we need to compute the corresponding values of g(x)g(x) for x=1,2,3,4,5x = 1, 2, 3, 4, 5.

Solution Steps:

  1. To calculate g(x)g(x), substitute xx into the equation g(x)=f(x+1)−2g(x) = f(x+1) - 2.

    • This means that for each xx, find the value of f(x+1)f(x+1) from the f(x)f(x) table.
    • Subtract 2 from f(x+1)f(x+1) to get g(x)g(x).

  2. Compute g(x)g(x) for each xx using the given data:

    • For x=1x = 1, g(1)=f(1+1)−2=f(2)−2=2−2=0g(1) = f(1+1) - 2 = f(2) - 2 = 2 - 2 = 0.
    • For x=2x = 2, g(2)=f(2+1)−2=f(3)−2=4−2=2g(2) = f(2+1) - 2 = f(3) - 2 = 4 - 2 = 2.
    • For x=3x = 3, g(3)=f(3+1)−2=f(4)−2=5−2=3g(3) = f(3+1) - 2 = f(4) - 2 = 5 - 2 = 3.
    • For x=4x = 4, g(4)=f(4+1)−2=f(5)−2=6−2=4g(4) = f(4+1) - 2 = f(5) - 2 = 6 - 2 = 4.
    • For x=5x = 5, g(5)=f(5+1)−2=f(6)−2=8−2=6g(5) = f(5+1) - 2 = f(6) - 2 = 8 - 2 = 6.

Completed Table:

\hline x & g(x) \\ \hline 1 & 0 \\ 2 & 2 \\ 3 & 3 \\ 4 & 4 \\ 5 & 6 \\ \hline \end{array}$$ ### Final Answer: The completed table for $$g(x)$$ is: $$\begin{array}{|c|c|} x & g(x) \\ 1 & 0 \\ 2 & 2 \\ 3 & 3 \\ 4 & 4 \\ 5 & 6 \\ \end{array}$$ Let me know if you need further details or an explanation! --- ### Questions for Further Exploration: 1. Can you describe the relationship between $$g(x)$$ and $$f(x)$$ in terms of a shift and subtraction? 2. What happens to $$g(x)$$ if we replace $$-2$$ with another constant $$c$$? 3. How would the table change if $$g(x) = f(x+2) - 2$$? 4. Could this relationship $$g(x) = f(x+1) - 2$$ be expressed graphically? 5. What is the domain and range of $$g(x)$$ given the provided $$f(x)$$? ### Tip: When working with function transformations, always carefully note how the input and output are modified based on the rule!

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Math Problem Analysis

Mathematical Concepts

Functions
Function Transformation
Algebra

Formulas

g(x) = f(x+1) - 2

Theorems

Function Transformation Theorem

Suitable Grade Level

Grades 8-10

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