Math Problem Statement
solve
Solution
Let's break down the problem step by step:
Given:
Part (a): Calculate .
1. Find :
To find the inverse of , we express , and solve for in terms of .
Thus, .
2. Find :
Using the inverse function:
3. Find :
For , the inverse is straightforward. Set , and solve for :
Thus, .
4. Find :
5. Multiply :
Now, multiply the two values:
So, the answer to part (a) is .
Part (b): Show that .
First, we need to calculate the composition , find its inverse, and then evaluate it at 3.
1. Find :
The composition means , so:
Thus, .
2. Find the inverse of :
Set , and solve for :
Thus, .
3. Evaluate :
Now substitute into the inverse function:
Notice that is the same as , so:
This proves the desired result.
Final Answers:
- Part (a):
- Part (b):
Would you like further details or clarifications on any part of the solution?
Here are 5 related questions:
- How do you generally find the inverse of an exponential function?
- What are the steps for composing two functions?
- Can you explain why worked out this way?
- How would you find the inverse of a more complex composite function?
- What is the significance of natural logarithms in inverse calculations?
Tip: Always check your composition when dealing with inverse functions to ensure that the result makes sense when plugging back into the original!
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Math Problem Analysis
Mathematical Concepts
Inverse functions
Exponential functions
Function composition
Formulas
Inverse of f(x) = e^(2x) is f^(-1)(x) = (1/2)ln(x)
Inverse of g(x) = x + 1 is g^(-1)(x) = x - 1
(f ∘ g)^(-1)(x) = (1/2)ln(x) - 1
Theorems
Properties of inverse functions
Logarithmic identities
Suitable Grade Level
Grades 11-12 (Advanced High School Mathematics or Pre-Calculus)
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