Problem 41 - Algebra for College Students

Question

(a) find \(f^{-1}\) and (b) verify that \(\left(f \circ f^{-1}\right)(x)=x\) and \(\left(f^{-1} \circ f\right)(x)=x\). $$ f(x)=\frac{3}{4} x-\frac{5}{6} $$

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = \frac{4}{3}x + \frac{10}{9}\) and it satisfies both \((f \circ f^{-1})(x) = x\) and \((f^{-1} \circ f)(x) = x\).

Step 1: Express the Function

Let's start with the given function: \( f(x) = \frac{3}{4}x - \frac{5}{6} \).

Step 2: Swap Variables

To find the inverse, replace \( f(x) \) with \( y \) to get \( y = \frac{3}{4}x - \frac{5}{6} \). Now swap \( x \) and \( y \): \( x = \frac{3}{4}y - \frac{5}{6} \).

Step 3: Solve for y

Now, solve for \( y \):1. Add \( \frac{5}{6} \) to both sides: \[ x + \frac{5}{6} = \frac{3}{4}y \]2. Multiply both sides by \( \frac{4}{3} \) to solve for \( y \): \[ y = \frac{4}{3} \left( x + \frac{5}{6} \right) \]3. Distribute \( \frac{4}{3} \): \[ y = \frac{4}{3}x + \frac{4}{3} \cdot \frac{5}{6} \]4. Simplify the expression: \[ y = \frac{4}{3}x + \frac{10}{9} \]

Step 4: Write the Inverse Function

The inverse function \( f^{-1}(x) \) is:\[ f^{-1}(x) = \frac{4}{3}x + \frac{10}{9} \]

Step 5: Verify \(f \circ f^{-1}(x) = x\)

To verify \( (f \circ f^{-1})(x) = x \), substitute \( f^{-1}(x) \) into \( f(x) \):1. \[ f \left( \frac{4}{3}x + \frac{10}{9} \right) \]2. Substitute into \( f(x) = \frac{3}{4}(\text{input}) - \frac{5}{6} \): \[ \frac{3}{4} \left( \frac{4}{3}x + \frac{10}{9} \right) - \frac{5}{6} \]3. Simplify: \[ x + \frac{10}{12} - \frac{5}{6} = x + \frac{10}{12} - \frac{10}{12} = x \]

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

To verify \( (f^{-1} \circ f)(x) = x \), substitute \( f(x) \) into \( f^{-1}(x) \):1. \[ f^{-1} \left( \frac{3}{4}x - \frac{5}{6} \right) \]2. Substitute into \( f^{-1}(x) = \frac{4}{3}(\text{input}) + \frac{10}{9} \): \[ \frac{4}{3} \left( \frac{3}{4}x - \frac{5}{6} \right) + \frac{10}{9} \]3. Simplify: \[ x - \frac{10}{9} + \frac{10}{9} = x \]

Key Concepts

Function Verification

Verifying an inverse function is a crucial step in understanding its validity. It ensures that the original function and its inverse can return to the initial input when composed with each other.

• First, calculate the composition of the function and its inverse, noted as \((f \circ f^{-1})(x)\).
• To successfully verify, this should simplify to \(x\).

Verification involves substituting the expression for \(f^{-1}(x)\) into \(f(x)\). This means you're replacing the inputs of the original function with the calculated inverse function. If after simplifying, the output is indeed \(x\), it proves that \(f(x)\) is the true inverse.
Next, interchange the roles by substituting \(f(x)\) into \(f^{-1}(x)\), noted as \((f^{-1} \circ f)(x)\). This composition should also yield \(x\). If both compositions lead back to the initial input \(x\), it confirms the functions are inverses.

Algebra

Algebra is the mathematical language that helps us find inverse functions. It's all about manipulating equations to express one variable in terms of another. When given a function, we use algebra to rearrange terms and solve for the inverse.

• First, we substitute \(f(x)\) with \(y\) to simplify the process.
• Next, we swap the variables. What was once an input is now an output and vice versa.
• The goal is to isolate \(y\), which becomes \(f^{-1}(x)\).

Doing this requires operations like addition, subtraction, multiplication, and division. Each step follows algebraic rules to maintain equation balance. For instance, adding the same number to both sides, or multiplying each side by the same factor. These operations help shift variables and isolate them, crucial in finding the inverse function.

Composition of Functions

Composition of functions is a method where one function is applied to the result of another function. It's often written as \((f \circ g)(x)\) and is read as \(f\) of \(g\) of \(x\).

• For verifying inverses, you use composition to cross check if \((f \circ f^{-1})(x) = x\) and \((f^{-1} \circ f)(x) = x\).
• This involves inserting one function into another. By performing this operation, you confirm if the two functions truly undo each other's actions.
• If both results equal \(x\), it highlights that each function serves as a mirror to the other. This reflection confirms they fit together perfectly within their domain and range.

Understanding this concept is the key to verifying inverse functions. When the compositions return to the original input, it endorses the relationship between a function and its inverse. This ensures the calculations were correctly performed and both functions correctly correspond to each other.