Math Problem Statement
Which point is on the graph of f^{-1}(x), the inverse function, given the following set of points on the graph of f(x): (2, 3), (3, 5), (4, 7), (5, 9), (6, 11), (7, 13)?
Solution
To find which point is on the graph of , the inverse function, we need to understand the relationship between the function and its inverse .
Given the set of points on the graph of :
x & y \\ \hline 2 & 3 \\ 3 & 5 \\ 4 & 7 \\ 5 & 9 \\ 6 & 11 \\ 7 & 13 \\ \end{array}$$ For the inverse function $$f^{-1}(x)$$, the roles of $$x$$ and $$y$$ are swapped. This means that if $$(a, b)$$ is on the graph of $$f(x)$$, then $$(b, a)$$ will be on the graph of $$f^{-1}(x)$$. Let's check the possible points provided: - Option A: $$(-3, -5)$$ — This point doesn't match any reverse pair from the given table. - Option B: $$(3, 5)$$ — This point is not the reverse of any given pair but rather already in the function table. - Option C: $$(-5, -3)$$ — This point also doesn't match any reverse pair from the table. - Option D: $$(5, 3)$$ — This point is the reverse of the pair $$(3, 5)$$, which is on the table. Thus, the correct point that is on the graph of $$f^{-1}(x)$$ is **(5, 3)**. Would you like more details or have any questions? --- Here are 5 related questions to deepen your understanding: 1. How do you determine the inverse of a function algebraically? 2. What are the properties of a function that guarantee its inverse will also be a function? 3. How can you graphically find the inverse of a function? 4. Why is the horizontal line test important when discussing inverses of functions? 5. What happens to the domain and range when you take the inverse of a function? **Tip:** When finding the inverse of a function, always ensure that the original function is one-to-one (passes the horizontal line test), which guarantees the inverse will be a valid function.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Inverse Functions
Graphing
Formulas
f^{-1}(x) means swapping the x and y values from f(x)
Theorems
If (a, b) is on the graph of f(x), then (b, a) is on the graph of f^{-1}(x).
Suitable Grade Level
Grades 9-12