Math Problem Statement
Evaluate each expression using the values given in the table.
Solution
To solve each of the given compositions using the table, let's evaluate each expression one by one. I’ll refer to and values from the table based on the given inputs.
Provided Table:
x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline f(x) & -4 & -3 & -2 & -1 & 2 & 3 & 4 \\ g(x) & 5 & 2 & 0 & -1 & 0 & 2 & 5 \\ \end{array}$$ ### Solutions #### a. $$(f \circ g)(1)$$ 1. First, find $$g(1)$$ from the table: $$g(1) = 0$$. 2. Now, find $$f(g(1)) = f(0)$$: $$f(0) = -1$$. 3. Therefore, $$(f \circ g)(1) = -1$$. #### b. $$(f \circ g)(-1)$$ 1. Find $$g(-1)$$: $$g(-1) = 0$$. 2. Then, find $$f(g(-1)) = f(0)$$: $$f(0) = -1$$. 3. So, $$(f \circ g)(-1) = -1$$. #### c. $$(g \circ f)(-1)$$ 1. Find $$f(-1)$$: $$f(-1) = -2$$. 2. Then, find $$g(f(-1)) = g(-2)$$: $$g(-2) = 2$$. 3. Thus, $$(g \circ f)(-1) = 2$$. #### d. $$(g \circ f)(0)$$ 1. Find $$f(0)$$: $$f(0) = -1$$. 2. Now, find $$g(f(0)) = g(-1)$$: $$g(-1) = 0$$. 3. Therefore, $$(g \circ f)(0) = 0$$. #### e. $$(g \circ g)(-2)$$ 1. Find $$g(-2)$$: $$g(-2) = 2$$. 2. Then, find $$g(g(-2)) = g(2)$$: $$g(2) = 0$$. 3. So, $$(g \circ g)(-2) = 0$$. #### f. $$(f \circ f)(-1)$$ 1. Find $$f(-1)$$: $$f(-1) = -2$$. 2. Now, find $$f(f(-1)) = f(-2)$$: $$f(-2) = -3$$. 3. Thus, $$(f \circ f)(-1) = -3$$. ### Summary of Answers - **a.** $$(f \circ g)(1) = -1$$ - **b.** $$(f \circ g)(-1) = -1$$ - **c.** $$(g \circ f)(-1) = 2$$ - **d.** $$(g \circ f)(0) = 0$$ - **e.** $$(g \circ g)(-2) = 0$$ - **f.** $$(f \circ f)(-1) = -3$$ Would you like more details on any specific part of this process? --- ### Related Questions: 1. How would you evaluate $$(f \circ g)(2)$$ using the same table? 2. Can you explain why composition functions like $$(g \circ f)(x)$$ can yield different results depending on the order? 3. What is the process for evaluating $$(f \circ f)(2)$$ with the given values? 4. How would changing values in the table affect the results of these compositions? 5. Could you determine $$(f \circ g)(3)$$ from the table? ### Tip: When working with function compositions, always start from the innermost function and work outward to avoid mistakes in evaluation.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Function Composition
Evaluation of Functions
Formulas
(f ∘ g)(x) = f(g(x))
(g ∘ f)(x) = g(f(x))
Theorems
Function Composition
Suitable Grade Level
Grades 9-11
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