Math Problem Statement
Use the Inverse Function Property to determine whether f and g are inverses of each other.
f(x) =
1/
(x − 10)
, x ≠ 10; g(x) =
1/x + 10
Solution
To determine whether and are inverses of each other, we can use the Inverse Function Property, which states:
If and are inverse functions, then: and
Step 1: Compute
Given , we substitute it into :
So, .
Step 2: Compute
Now, substitute into :
So, .
Conclusion:
Since both and , we can conclude that and are inverses of each other.
Would you like further details or have any questions?
Here are 5 related questions to expand your understanding:
- What are the general conditions for two functions to be inverses of each other?
- How can you visually verify if two functions are inverses using their graphs?
- How does the domain and range of a function affect its inverse?
- Can all functions have an inverse? Why or why not?
- How does composition of functions help in verifying if two functions are inverses?
Tip: Inverse functions "undo" each other, meaning the output of one function becomes the input of the other, and vice versa.
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Math Problem Analysis
Mathematical Concepts
Functions
Inverse Functions
Composition of Functions
Formulas
(f ∘ g)(x) = f(g(x))
(g ∘ f)(x) = g(f(x))
Theorems
Inverse Function Property
Suitable Grade Level
Grades 10-12
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