Problem 11 - Contemporary Precalculus

Question

Use algebra to find the inverse of the given one-to-one function. $$f(x)=5 x-4$$

Step-by-Step Solution

Answer

Question: Find the inverse of the one-to-one function $$f(x) = 5x - 4$$. Answer: The inverse function of $$f(x) = 5x - 4$$ is $$f^{-1}(x) = \frac{x + 4}{5}$$.

Step 1: Replace function notation with a variable

Instead of using function notation (f(x)), we will use a variable, y, to represent the function's output. So the equation becomes: $$y = 5x - 4$$

Step 2: Swap the roles of x and y

We are going to replace y with x and x with y to obtain the inverse function. This gives us the equation: $$x = 5y - 4$$

Step 3: Solve for y to obtain the inverse function

Now we need to solve for y in terms of x. Follow these steps: 1. Add 4 to both sides of the equation: $$x + 4 = 5y$$ 2. Divide both sides of the equation by 5: $$\frac{x + 4}{5} = y$$ So the inverse function is: $$f^{-1}(x) = \frac{x + 4}{5}$$

Key Concepts

One-to-One Function

In the world of mathematics, a **One-to-One Function** is a function where exactly one x-value corresponds to exactly one y-value. This is important because only one-to-one functions have inverses that are also functions.Identifying whether a function is one-to-one can be done using the horizontal line test:- If any horizontal line intersects the graph of the function more than once, the function is not one-to-one.Understanding that the function \(f(x) = 5x - 4\) is one-to-one exposes a critical element: its inverse \(f^{-1}(x)\) is also a function. When you are given such a function, and you're asked to find its inverse, rest assured, you’re dealing with numbers that consistently follow this single-pair relationship.

Algebraic Manipulation

Algebraic Manipulation is a process that essentially involves rearranging an equation to solve for one of its variables. In finding the inverse of a function, algebraic manipulation is used to isolate one variable from the function.Let's break down the process in finding the inverse of \(f(x) = 5x - 4\):- **Step 1**: Replace \(f(x)\) with \(y\) to ease manipulation. The equation becomes \(y = 5x - 4\).
- **Step 2**: Swap \(x\) and \(y\). This is where you switch their places to set the foundation for the inverse: \(x = 5y - 4\).
- **Step 3**: Solve for \(y\) using algebraic rearrangements: - Add 4 to each side: \(x + 4 = 5y\). - Divide every term by 5 to isolate \(y\): \(y = \frac{x + 4}{5}\).
These steps uncover the inverse function, expressing how effectively algebraic rules can be used to find solutions.

Function Notation

**Function Notation** provides a way to represent functions and their inversions without ambiguity. Rather than writing cumbersome equations, function notation communicates mathematical ideas succinctly.For example, in the function \(f(x) = 5x - 4\), \(f(x)\) signifies the output of the function when input \(x\) is applied.
When finding an inverse, the goal is to reverse this process. The inverse function notation, \(f^{-1}(x)\), provides a clear indication that we’re discussing the reverse operation. The derived expression \(f^{-1}(x) = \frac{x+4}{5}\) conveys that if \(x\) is plugged into the inverse function, the original input for \(f(x)\) is retrieved.**Key Points to Remember:**- Function notation simplifies the relationship between \(x\) and \(f(x)\).- It ensures clarity when discussing functions and their inverses, crucial when performing operations such as differentiation or integration.