Problem 28 - Thomas Calculus in SI Units

Question

Gives a formula for a function \(y=f(x) .\) In each case, find \(f^{-1}(x)\) and identify the domain and range of \(f^{-1} .\) As a check, show that \(f\left(f^{-1}(x)\right)=f^{-1}(f(x))=x\). $$f(x)=(1 / 2) x-7 / 2$$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = 2x + 7 \) with domain and range \( (-\infty, \infty) \).

Step 1: Express the function equation

Given the function \( f(x) = \frac{1}{2}x - \frac{7}{2} \). We need to solve for the inverse function \( f^{-1}(x) \). Start by replacing \( f(x) \) with \( y \):\[ y = \frac{1}{2}x - \frac{7}{2} \]

Step 2: Swap and Solve for x

Swap \( x \) and \( y \) in the equation to find the inverse:\[ x = \frac{1}{2}y - \frac{7}{2} \]Now solve for \( y \) to express the inverse function.

Step 3: Solve for y

Add \( \frac{7}{2} \) to both sides to isolate the term with \( y \):\[ x + \frac{7}{2} = \frac{1}{2}y \]Multiply every term by 2 to solve for \( y \):\[ 2x + 7 = y \]Therefore, the inverse function is:\[ f^{-1}(x) = 2x + 7 \]

Step 4: Determine the domain and range of the inverse function

For the original function \( f(x) = \frac{1}{2}x - \frac{7}{2} \), the domain is all real numbers \(( -\infty, \infty )\). Since the domain of \( f \) becomes the range of \( f^{-1} \), and vice versa, the domain of \( f^{-1}(x) = 2x + 7 \) is \( ( -\infty, \infty )\), and its range is also \( ( -\infty, \infty )\).

Step 5: Verify the Inverse Function

To verify \( f(f^{-1}(x)) = x \), substitute \( 2x + 7 \) into \( f \):\[ f(f^{-1}(x)) = f(2x + 7) = \frac{1}{2}(2x + 7) - \frac{7}{2} \]Simplify:\[ = x + \frac{7}{2} - \frac{7}{2} = x \]Similarly, verify \( f^{-1}(f(x)) = x \) by substituting \( f(x) \):\[ f^{-1}\left(\frac{1}{2} x - \frac{7}{2}\right) = 2\left(\frac{1}{2} x - \frac{7}{2}\right) + 7 \]Simplify:\[ = x - 7 + 7 = x \]Both conditions are satisfied, confirming the calculation.

Key Concepts

Domain and Range

When dealing with functions and their inverses, understanding the concepts of domain and range is crucial.
The domain of a function is the complete set of possible input values (usually represented as \(x\)) which result in a valid output from the function. For the function \(f(x) = \frac{1}{2}x - \frac{7}{2}\), the domain is all real numbers, written as \[(-\infty, \infty)\].
Similarly, the range of a function is the complete set of possible output values. This means any value that the function can produce as \(y = f(x)\). Since our given function is linear, like a straight line, it can also produce all real numbers, so the range is \[(-\infty, \infty)\].
For the inverse function \(f^{-1}(x)\), the roles of domain and range are reversed. The domain of the original function becomes the range of the inverse and vice versa. Therefore, \(f^{-1}(x) = 2x + 7\) has a domain of \[(-\infty, \infty)\] and a range of \[(-\infty, \infty)\]. These concepts help ensure that functions and their inverses are well-defined and consistent when swapped.

Function Verification

One vital step in working with inverse functions is to verify that you have correctly found the inverse. This involves confirming two conditions:

• \(f(f^{-1}(x)) = x\)
• \(f^{-1}(f(x)) = x\)

You need to check these both because they essentially serve as a double-check that your computations are correct.
To verify \(f(f^{-1}(x)) = x\), we substitute the inverse function into the original function. For this exercise, we take \(f^{-1}(x) = 2x + 7\) and substitute it into \(f\):
\[f(f^{-1}(x)) = f(2x + 7) = \frac{1}{2}(2x + 7) - \frac{7}{2} = x + \frac{7}{2} - \frac{7}{2} = x\]
Similarly, for \(f^{-1}(f(x)) = x\), substitute the original function into the inverse:
\[f^{-1}\left(\frac{1}{2} x - \frac{7}{2}\right) = 2\left(\frac{1}{2} x - \frac{7}{2}\right) + 7 = x - 7 + 7 = x\]
These statements confirm that both the function and its inverse undo each other, which must always be the case for a proper inverse.

Steps to Find Inverse Function

Finding the inverse function involves a series of clear and straightforward steps. Let's walk through these steps using the function \(f(x) = \frac{1}{2}x - \frac{7}{2}\):

• **Step 1: Express the Function Equation**
  Start by writing the function in equation form, replacing \(f(x)\) with \(y\). This gives \(y = \frac{1}{2}x - \frac{7}{2}\).


• **Step 2: Swap and Solve for \(x\)**
  Swap the variables \(x\) and \(y\). This converts the equation into \(x = \frac{1}{2}y - \frac{7}{2}\). The goal is now to solve this equation for \(y\).


• **Step 3: Solve for \(y\)**
  Add \(\frac{7}{2}\) to both sides to get \(x + \frac{7}{2} = \frac{1}{2}y\). Next, multiply every term by 2 to isolate \(y\), resulting in \(2x + 7 = y\).

The inverse function is therefore \(f^{-1}(x) = 2x + 7\). By following these structured steps, you ensure a systematic approach to finding the inverse of any function.