Math Problem Statement
Solution
Let's analyze the given tables to determine the correct expressions for and in terms of .
For :
The table for is:
x & -2 & -1 & 0 & 1 & 2 \\ \hline f(x) & 1 & -4 & 4 & -3 & 2 \\ \end{array}$$ The table for $$g(x)$$ is: $$\begin{array}{c|c|c|c|c|c} x & -3 & -2 & -1 & 0 & 1 \\ \hline g(x) & 1 & -4 & 4 & -3 & 2 \\ \end{array}$$ Notice that $$g(x) = f(x+1)$$. This can be verified by substituting values: - For $$x = -3$$, $$g(-3) = 1$$ and $$f(-3 + 1) = f(-2) = 1$$. - For $$x = -2$$, $$g(-2) = -4$$ and $$f(-2 + 1) = f(-1) = -4$$. - And so on for all other values. Thus, the correct answer is $$g(x) = f(x+1)$$. ### For $$h(x)$$: The table for $$h(x)$$ is: $$\begin{array}{c|c|c|c|c|c} x & -2 & -1 & 0 & 1 & 2 \\ \hline h(x) & 2 & -3 & 5 & -2 & 3 \\ \end{array}$$ To determine $$h(x)$$, we compare it with the table for $$f(x)$$: - For $$x = -2$$, $$h(-2) = 2$$ and $$f(-2 - 1) = f(-3)$$, which isn't available, but checking other values we see that $$h(x) = f(x-1)$$. - For $$x = -1$$, $$h(-1) = -3$$ and $$f(-1-1) = f(-2) = 1$$. Hence, the correct answer is $$h(x) = f(x-1)$$. Would you like more details, or have any questions about these solutions? Here are 5 related questions to further explore this topic: 1. How can we derive a function in terms of another when given tabular data? 2. What transformations can we apply to functions to shift them horizontally or vertically? 3. How do different function notations like $$f(x+1)$$ or $$f(x-1)$$ affect the graph of the function? 4. Can we determine the inverse of a function from its table of values? 5. What are some common mistakes to avoid when matching tables to function expressions? **Tip:** When comparing functions, always check for patterns in both the input (x-values) and the output (f(x) values).Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Function transformation
Table analysis
Formulas
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Theorems
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Suitable Grade Level
High School