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Chapter 3: Problem 32

Find \(f^{-1}(x) .\) State any restrictions on the domain of \(f^{-1}(x)\) $$f(x)=-x+3$$

Short Answer

Expert verified
\The inverse function \(f^{-1}(x)\) is \(f^{-1}(x) = 3 - x\). And there are no restrictions on the domain of \(f^{-1}(x)\) as \(x\) can be any real number.

Step by step solution

01

Swap \(x\) and \(y\)

Start by replacing \(f(x)\) with \(y\). This gives us \(y = -x + 3\). Then, we swap \(x\) and \(y\) which gives us \(x = -y + 3\).
02

Solve for \(y\)

To isolate \(y\), add \(y\) to both sides and substract \(x\) from both sides: \(x + y = 3 \Rightarrow y = 3 - x\).
03

Transform \(y\) into \(f^{-1}(x)\)

Now we can replace \(y\) with \(f^{-1}(x)\) this gives us \(f^{-1}(x) = 3 - x\).
04

Determine the domain of \(f^{-1}(x)\)

For a linear function like the one we have here, there are no restrictions for \(x\), because \(x\) can be any real number. Therefore, there are no restrictions on the domain of \(f^{-1}(x)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Domain of Inverse Functions
The domain of an inverse function plays a crucial role in understanding its behavior. To find the domain of an inverse function, consider the range of the original function. This is because the range of the original function becomes the domain of the inverse function. In the exercise given, we are dealing with the function \( f(x) = -x + 3 \). As a linear function, \( f(x) \) covers all real numbers because linear functions have an unrestricted range. This implies that the inverse function \( f^{-1}(x) \) is also defined for all real numbers. Thus, the domain of \( f^{-1}(x) \) is \((-\infty, \infty)\), meaning any real number can be an input.
  • Remember, to find an inverse function, you swap "x" and "y" and then solve for "y" to get the inverse.
  • Always confirm the domain by checking if the original function's range covers all real numbers.
  • Linear functions' inverse functions generally have no domain restrictions.
Linear Functions
Linear functions are one of the simplest and most straightforward types of functions in mathematics. They are represented in the form \( f(x) = mx + b \), where \( m \) and \( b \) are constants. The 'm' represents the slope of the line, and 'b' represents the y-intercept. In our case, the exercise involves the function \( f(x) = -x + 3 \), indicating:
  • The slope \(m\) is \(-1\), meaning the line decreases as you move from left to right.
  • The y-intercept \(b\) is \(3\), meaning the line crosses the y-axis at point \( (0, 3) \).
Linear functions continue indefinitely in both directions along the x-axis and their simplicity makes them easy to work with when determining inverses.
  • The inverse of a linear function is also a linear function.
  • Simplicity of these functions allows for clear domain and range boundaries; both typically span all real numbers.
Function Composition
Function composition is the process of combining two functions where one function is applied to the results of another. It's an essential concept for understanding how to verify inverse functions. Typically, if you have two functions, \( f \) and \( g \), the composition is written as \( (f \circ g)(x) \), which means apply \( g \) first, then \( f \).An important property of inverse functions is that composing a function with its inverse yields the identity function. For instance, \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \). This characteristic confirms that \( f(x) \) and \( f^{-1}(x) \) truly are inverses of one another.
  • To verify if two functions are inverses, compose them. The composition should result in \(x\).
  • Ensure proper order: the inner function is evaluated first in compositions.
  • Function compositions can test functionality and provide insights into the domain and range relationships.

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