Math Problem Statement
Solution
Let's break down the steps needed to solve this problem set:
Given:
(a) Show that
This step requires simplifying .
-
Step 1: Simplify .
- Factorize the quadratic denominator if possible.
-
Step 2: Combine the two terms of into one expression.
(b) Find
To find the inverse function of , follow these steps:
- Set .
- Swap and to find the inverse.
- Solve for in terms of .
(c) Find the domain of
Since the inverse function is , find the domain by considering the range of the original function .
(d) Find and its domain
Since , follow these steps to find its inverse:
- Set .
- Solve for in terms of .
The domain of corresponds to the range of .
(e) Sketch the curves of and
- Plot the logarithmic curve .
- Plot the inverse curve.
- Identify the points where the curves cross the axes.
(f) Solve
- Use the composite function .
- Set and solve for in terms of .
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Math Problem Analysis
Mathematical Concepts
Algebra
Functions
Inverse Functions
Logarithmic Functions
Composite Functions
Formulas
f(x) = 1/(2x - 1)
g(x) = ln(x + 1)
f^(-1)(x) and g^(-1)(x)
Theorems
Inverse Function Theorem
Logarithmic Properties
Suitable Grade Level
Grades 11-12 (Pre-Calculus or Calculus)
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