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Chapter 1: Problem 23

a. Find the imerse of each function (on the given interval, if specified) and write it in the form \(y=f^{-1}(x)\). b. Verify the relationships \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). $$f(x)=6-4 x$$

Short Answer

Expert verified
Based on the given function \(f(x)=6-4x\), we've found its inverse function as \(f^{-1}(x)=\frac{1}{4}(6-x)\). We also verified the relationships \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\), which confirms the correctness of the inverse function and its relationships with the original function.

Step by step solution

01

Find the inverse function \(f^{-1}(x)\)

To find the inverse function, we need to solve the equation \(f(x)=y\) for \(x\). Here, \(f(x)=6-4x=y\). So, we have: $$ y=6-4x $$ Now, we solve for \(x\): $$ 4x=6-y $$ $$ x=\frac{6-y}{4} =\frac{1}{4}(6-y) $$ Now, we replace \(y\) with \(f^{-1}(x)\) and \(x\) with \(y\): $$ f^{-1}(x)=\frac{1}{4}(6-x) $$
02

Verify the relationship \(f\left(f^{-1}(x)\right)=x\)

To verify this relationship, we need to plug \(f^{-1}(x)\) into \(f(x)\): $$ f\left(f^{-1}(x)\right)=6-4\left(\frac{1}{4}(6-x)\right) $$ Now, we simplify: $$ f\left(f^{-1}(x)\right)=6-(6-x) $$ $$ f\left(f^{-1}(x)\right)=x $$ So, the relationship \(f\left(f^{-1}(x)\right)=x\) is true.
03

Verify the relationship \(f^{-1}(f(x))=x\)

To verify this relationship, we need to plug \(f(x)\) into \(f^{-1}(x)\): $$ f^{-1}(f(x))=f^{-1}(6-4x) $$ Now, we substitute the expression for \(f^{-1}(x)\): $$ f^{-1}(6-4x)=\frac{1}{4}(6-(6-4x)) $$ Now, we simplify: $$ f^{-1}(6-4x)=\frac{1}{4}(4x) $$ $$ f^{-1}(f(x))=x $$ So, the relationship \(f^{-1}(f(x))=x\) is true. Thus, we have found the inverse function \(f^{-1}(x)=\frac{1}{4}(6-x)\) and verified the relationships \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\).

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Key Concepts

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

Function Verification
Verifying the correctness of an inverse function is crucial to ensure that it perfectly reverses the operations of the original function. In other words, the goal is to confirm that applying the inverse function will bring you back to your starting value. This verification process is grounded in two core relationships:
  • \( f(f^{-1}(x)) = x \)
  • \( f^{-1}(f(x)) = x \)
The essence of function verification lies in checking these two conditions. For the given function \( f(x) = 6 - 4x \) and its inverse \( f^{-1}(x) = \frac{1}{4}(6-x) \), we satisfy these conditions via substitution and simplification. By plugging \( f^{-1}(x) \) back into \( f(x) \), and vice-versa, you need to arrive back at \( x \), thus confirming a well-defined inverse.
Algebraic Manipulation
Algebraic manipulation is the process of rearranging and simplifying equations to find solutions. When we seek an inverse function, it involves these careful steps:
  • Start with the inverse equation \( y = f(x) \).
  • Solve for \( x \) in terms of \( y \). This step often involves isolating the desired variable through operations such as addition, subtraction, multiplication, or division.
  • Swap the roles of \( x \) and \( y \) after solving, resulting in \( y = f^{-1}(x) \).
For the function \( f(x) = 6 - 4x \), solving for \( x \) gave us \( x = \frac{1}{4}(6-y) \). By swapping variables, the inverse equation becomes \( f^{-1}(x) = \frac{1}{4}(6-x) \). Mastery in these manipulations is critical for accurately finding inverses and verifying them.
Mathematical Proof
Proving mathematical concepts holds everything together in mathematics. By providing a structured argument, mathematical proof offers validation and confidence in the solutions we derive. To prove that \( f(x) = 6 - 4x \) and \( f^{-1}(x) = \frac{1}{4}(6-x) \) are indeed inverses, we show:
  • \( f(f^{-1}(x)) = x \)
  • \( f^{-1}(f(x)) = x \)
These proofs rely on substituting the inverse function into its original, and vice-versa, simplified step by step:
  • Substituting \( f^{-1}(x) \) into \( f(x) \) results in \( f(f^{-1}(x)) = 6 - 4(\frac{1}{4}(6-x)) = x \), thereby verifying this relationship.
  • Similarly, substituting \( f(x) \) into \( f^{-1}(x) \) leads to \( f^{-1}(f(x)) = \frac{1}{4}(6 - (6 - 4x)) = x \).
This thorough verification through substitution assures that we've proven the inverse function relationship accurately. Approaching such proofs encourages a deep understanding of inverse functions and strengthens mathematical reasoning skills.

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