Verified Answer :
- Confirm the function \(f(x) = \frac{3x - 8}{7 + 3x}\) is invertible by verifying the determinant condition \(ad-bc eq 0\).
- Set \(y = f(x)\) and swap variables to form the equation \(x = \frac{3y - 8}{7 + 3y}\).
- Isolate \(y\) by expanding and rearranging the equation to solve for the inverse in terms of \(x\).
- State the final inverse function as \(f^{-1}(x) = \frac{7x + 8}{3 - 3x}\).
- Final Answer: \(\boxed{f^{-1}(x) = \frac{7x + 8}{3 - 3x}}\)
Explanation
- Analysis of the Function
The function provided is \(f(x) = \frac{3x - 8}{7 + 3x}\). To find its inverse, we must first ensure that the function is invertible. A rational function of the form \(\frac{ax+b}{cx+d}\) is invertible if \(ad - bc eq 0\). In our case, \(a=3\), \(b=-8\), \(c=3\), and \(d=7\). Evaluating the determinant gives \(ad - bc = (3)(7) - (-8)(3) = 21 + 24 = 45\). Since \(45 eq 0\), the function is indeed one-to-one and has a well-defined inverse.
- Setting Up the Inverse Equation
To find the inverse function \(f^{-1}(x)\), we start by setting \(y = f(x)\), which gives:
\[y = \frac{3x - 8}{7 + 3x}\]
Next, we swap the roles of \(x\) and \(y\) to solve for the inverse relationship:
\[x = \frac{3y - 8}{7 + 3y}\]
Our goal is to isolate \(y\) in terms of \(x\).
- Solving for the Inverse Variable
Multiply both sides by the denominator \((7 + 3y)\) to clear it:
\[x(7 + 3y) = 3y - 8\]
Distribute \(x\) on the left side:
\[7x + 3xy = 3y - 8\]
Now, group all terms containing \(y\) on one side and the remaining terms on the other side:
\[3xy - 3y = -7x - 8\]
Factor out \(y\) from the left side:
\[y(3x - 3) = -(7x + 8)\]
Divide by \((3x - 3)\) to solve for \(y\):
\[y = \frac{-(7x + 8)}{3x - 3}\]
To simplify, we can multiply the numerator and denominator by \(-1\):
\[y = \frac{7x + 8}{3 - 3x}\]
Thus, the inverse function is \(f^{-1}(x) = \frac{7x + 8}{3 - 3x}\).
Examples
In data science and cryptography, inverse functions are essential for reversing operations. For example, if a signal is encoded through a transformation, the receiving system must apply the inverse transformation to recover the original information. Similarly, in economics, if a function maps a price to the quantity demanded, its inverse maps the quantity back to the price, allowing businesses to adjust pricing strategies based on target sales volumes. Understanding how to calculate inverse functions provides the foundational tools to reverse complex processes across various fields.