Problem 28 - Intermediate Algebra

Question

Find the inverse of the given function by using the "undoing process," and then verify that \(\left(f \circ f^{-1}\right)(x)=x\) and \(\left(f^{-1} \circ f\right)(x)=x\). (Objective 4) $$f(x)=\frac{3}{4} x-2$$

Step-by-Step Solution

Answer

The inverse of \( f(x) = \frac{3}{4}x - 2 \) is \( f^{-1}(x) = \frac{4}{3}x + \frac{8}{3} \).

Step 1: Replace f(x) with y

To find the inverse, start by replacing the function notation \( f(x) \) with \( y \). This gives us the equation \( y = \frac{3}{4}x - 2 \).

Step 2: Swap x and y

To find the inverse equation, swap \( x \) and \( y \). This means we rewrite the equation as \( x = \frac{3}{4}y - 2 \).

Step 3: Solve for y

Now, solve for \( y \) to get the inverse function. First, add 2 to both sides to get \( x + 2 = \frac{3}{4}y \).

Step 4: Multiply by the reciprocal

Multiply both sides by the reciprocal of \( \frac{3}{4} \), which is \( \frac{4}{3} \). This gives us \( y = \frac{4}{3}(x + 2) \).

Step 5: Simplify the inverse equation

Distribute \( \frac{4}{3} \) to both terms inside the parentheses to simplify: \( y = \frac{4}{3}x + \frac{8}{3} \). Thus, the inverse function \( f^{-1}(x) \) is \( \frac{4}{3}x + \frac{8}{3} \).

Step 6: Verify \(\left(f \circ f^{-1}\right)(x) = x\)

Substitute \( f^{-1}(x) = \frac{4}{3}x + \frac{8}{3} \) into \( f(x) \). Start with \( f\left(\frac{4}{3}x + \frac{8}{3}\right) = \frac{3}{4}\left(\frac{4}{3}x + \frac{8}{3}\right) - 2 \). Simplify to verify \( x \). After simplification, we indeed find \( x \).

Step 7: Verify \(\left(f^{-1} \circ f\right)(x) = x\)

Substitute \( f(x) = \frac{3}{4}x - 2 \) into \( f^{-1}(x) \). Start with \( f^{-1}\left(\frac{3}{4}x - 2\right) = \frac{4}{3}\left(\frac{3}{4}x - 2 + 2\right) \). Simplification shows this reduces to \( x \).

Key Concepts

Algebra

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. It's all about understanding relationships. In algebra, we typically use equations to define a variety of mathematical problems and solve for unknown values. This concept forms the base for understanding functions and their inverses.

Finding an inverse function is like unraveling the original equation. It involves working backwards to express the input variable in terms of the output. For the given function, \(f(x)=\frac{3}{4}x-2\), finding its inverse requires manipulating the algebraic expression. Start by replacing \(f(x)\) with \(y\), leading to \(y=\frac{3}{4}x-2\).

Then, we swap \(x\) and \(y\), yielding the equation \(x=\frac{3}{4}y-2\). Solving for \(y\) involves isolating it on one side. We need to undo every operation that manipulates \(y\). Start by adding 2 on each side to get \(x+2=\frac{3}{4}y\), and solve it by multiplying each side by the reciprocal of \(\frac{3}{4}\), which is \(\frac{4}{3}\). This step-by-step manipulation, typical in algebra, eventually reveals the inverse \(f^{-1}(x)=\frac{4}{3}x+\frac{8}{3}\).

Function Composition

Function composition involves creating a new function by applying one function to another. It's denoted by \((f \circ g)(x) = f(g(x))\), which means \(g(x)\) is placed inside of \(f(x)\).

When dealing with inverse functions, composition is vital in verifying if two functions are indeed inverses. For the function \(f(x)\) and its inverse \(f^{-1}(x)\), we check if their compositions result in the identity function \(x\). This means:

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

In our example: 1. Substitute \(f^{-1}(x) = \frac{4}{3}x + \frac{8}{3}\) into \(f(x)\) to form \(f(f^{-1}(x))\). Simplify it: \(f(\frac{4}{3}x + \frac{8}{3}) = \frac{3}{4}(\frac{4}{3}x + \frac{8}{3}) - 2\). Here, simplification should lead to \(x\).
2. Similarly, place \(f(x) = \frac{3}{4}x - 2\) into \(f^{-1}(x)\), then simplify \(f^{-1}(f(x))\) so that it also results in \(x\). This process helps ensure the discovered inverse is correct.

Verification of Inverses

Verification of inverse functions is crucial for confirming the relationship between a function and its inverse. To verify, you must perform function composition with both the original function and its inverse, then check whether they simplify back to the identity function, \(x\).

In our specific case, we have verified these steps:

• By substituting the inverse \( f^{-1}(x) = \frac{4}{3}x + \frac{8}{3}\) into \(f\) and simplifying, it must result in \(x\). This was proven by substitution and simplification: \(f(f^{-1}(x)) = \frac{3}{4}(\frac{4}{3}x + \frac{8}{3}) - 2 = x\).
• Likewise, insert \( f(x) = \frac{3}{4}x - 2\) into the inverse function and simplify \(f^{-1}(f(x))\). Confirm it also collapses to \(x\). This mutual simplification confirms correct inverses.

These verifications prove that we've correctly identified the inverse function. It's a rigorous practice ensuring that the processes we follow in algebra are both accurate and dependable. Verifying inverses not only confirms their correctness but enhances our comprehensive understanding of functions in algebra.