Math Problem Statement
Solution
The problem shown in the image is asking to prove algebraically that the functions and are inverse functions. To do this, we need to show that and .
Let's begin by evaluating and step by step.
1. Prove :
We are given:
Now, substitute into : [ f(g(x)) = f\left( \sqrt[3]{2x} \right) = \frac{\left( \sqrt[3]{2x} \right)^3}{2} ]
Simplify:
Thus, , which is correct.
2. Prove :
Now, substitute into :
Simplify:
Thus, , which is also correct.
Conclusion:
Both conditions and are satisfied, proving that and are indeed inverse functions.
Do you want more details or explanations on this? Here are 5 related questions to explore:
- What is the graphical interpretation of inverse functions?
- How do you find the inverse of a function in general?
- Can a function have more than one inverse?
- What are the conditions for two functions to be inverses?
- How can inverse functions be verified numerically?
Tip: Always check both and when proving functions are inverses.
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Math Problem Analysis
Mathematical Concepts
Algebra
Inverse Functions
Function Composition
Formulas
f(g(x)) = x
g(f(x)) = x
Theorems
Inverse function property
Suitable Grade Level
Grades 10-12
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