Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 10: Problem 31
Determine whether \(f\) and \(g\) are inverse functions. $$ f(x)=x \text { and } g(x)=\frac{1}{x} $$
Short Answer
Expert verified
No, \( f \) and \( g \) are not inverse functions.
Step by step solution
01
Understand the Definition of Inverse Functions
Two functions, \( f \) and \( g \), are inverse functions if their compositions yield the identity function, i.e., \( f(g(x)) = x \) and \( g(f(x)) = x \) for all \( x \) in the domains of the respective compositions.
02
Compose \( f(g(x)) \)
We substitute \( g(x) \) into \( f(x) \), to find \( f(g(x)) \). Since \( g(x) = \frac{1}{x} \) and \( f(x) = x \), we have:\[ f(g(x)) = f\left(\frac{1}{x}\right) = \frac{1}{x}. \] This should equal \( x \) if \( f \) and \( g \) are inverse functions.
03
Compose \( g(f(x)) \)
Next, substitute \( f(x) \) into \( g(x) \), to find \( g(f(x)) \). Since \( f(x) = x \) and \( g(x) = \frac{1}{x} \), we have:\[ g(f(x)) = g(x) = \frac{1}{x}. \] This should equal \( x \) if \( f \) and \( g \) are inverse functions.
04
Analyze the Results
From Step 2, we have \( f(g(x)) = \frac{1}{x} \) instead of \( x \). From Step 3, \( g(f(x)) = \frac{1}{x} \) rather than \( x \). Therefore, neither composition results in the identity function on the respective domains.
05
Conclusion
Since neither \( f(g(x)) \) nor \( g(f(x)) \) equals \( x \), \( f(x) = x \) and \( g(x) = \frac{1}{x} \) are not inverse functions of each other.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Function Composition
Function composition refers to combining two functions to form a new function. Essentially, you substitute one function into another. This is denoted by \( f(g(x)) \) or \( g(f(x)) \), meaning you take the output of function \( g \) and use it as the input for function \( f \), and vice versa. Function composition helps us determine how one function interacts with another.
- When attempting to find if two functions, say \( f \) and \( g \), are inverse to each other, you must check the compositions \( f(g(x)) \) and \( g(f(x)) \).
- Neither of these operations should change the input value \( x \) if the functions are true inverses.
Identity Function
The identity function is a critical concept when determining inverse functions. It is written as \( I(x) = x \). This function always returns its input as the output, essentially leaving \( x \) unchanged. If a function composed with its potential inverse returns the identity function, they are considered inverses.
- For inverses, function compositions like \( f(g(x)) = x \) and \( g(f(x)) = x \) must hold true. This means that after performing the composition, you end up with the identity function.
Domain of a Function
Understanding the domain of a function is crucial in evaluating function compositions and checking if functions are inverses. The domain of a function is the set of input values for which the function is defined. In simple terms, it's the collection of all \( x \) values you can plug into the function without causing any issues like division by zero.
- When evaluating function compositions for inverses, you must consider the domains of \( f \) and \( g \) to ensure that the compositions are valid. This is because an invalid input (like zero in a denominator) would break the function.
- For \( g(x) = \frac{1}{x} \), you must exclude \( x = 0 \) from its domain because the function would be undefined at that point.
