Problem 16 - Contemporary Precalculus

Question

Use algebra to find the inverse of the given one-to-one function. $$f(x)=5+\sqrt{3 x-2}$$

Step-by-Step Solution

Answer

Answer: The inverse function is $$f^{-1}(x)=\frac{1}{3}(x^2-10x+27)$$

Step 1: Replace f(x) with y

We replace the given function $$f(x)$$ with $$y$$: $$y=5+\sqrt{3x-2}$$

Step 2: Swap x and y

Now, we swap $$x$$ and $$y$$ in the equation: $$x=5+\sqrt{3y-2}$$

Step 3: Solve for y

We will solve the equation for $$y$$ by first isolating the square root term: Subtract 5 from both sides of the equation: $$x-5=\sqrt{3y-2}$$ Now, square both sides to get rid of the square root: $$(x-5)^2=(\sqrt{3y-2})^2$$ This simplifies to: $$x^2-10x+25=3y-2$$ Now, isolate $$y$$ by adding $$2$$ and then dividing by $$3$$: $$3y=x^2-10x+27$$ $$y=\frac{1}{3}(x^2-10x+27)$$

Step 4: Write the inverse function

Finally, we write the inverse function using the notation $$f^{-1}(x)$$: $$f^{-1}(x)=\frac{1}{3}(x^2-10x+27)$$

Key Concepts

Algebra

Algebra is like a toolbox for solving equations and manipulating expressions. It allows us to work with variables and constants to uncover values that make equations true. For example, when finding the inverse of a function, we utilize algebra to rearrange the equation to isolate the desired variable.

• We replace function notation like \( f(x) \) with a simple \( y \) to make manipulations easier.
• Next, we switch the places of \( x \) and \( y \), essentially reversing the roles. This helps us focus on finding the inverse.
• Then, using algebraic techniques such as addition, subtraction, squaring, or factoring, we solve for \( y \).
• Finally, we express \( y \) back in terms of \( x \), resulting in the inverse function.

By using these steps, algebra helps us understand how changing one variable affects another, even in complex functions. It’s a bit like a puzzle, but once you know where each piece fits, the solution becomes clear.

One-to-One Functions

A one-to-one function is special because it has a unique output for every unique input. This characteristic is crucial for finding inverse functions. If a function is not one-to-one, it can’t have a proper inverse.

• Think of it as a magical machine where no two different things become the same result. In mathematics, this means that if \( f(a) = f(b) \), then \( a \) must equal \( b \).
• This uniqueness allows us to flip the function around and find its inverse, which essentially "undoes" the original function.
• For example, if you pass a number through a one-to-one function and then through its inverse, you’ll end up right where you started.

Understanding whether a function is one-to-one helps us know if an inverse can exist. When dealing with inverse problems, it’s the foundational step to ensure our work is valid.

Function Notation

Function notation is a way of representing functions to easily identify inputs and outputs. It uses symbols like \( f(x) \) to indicate a function named \( f \) with input \( x \). This notation provides clarity when writing and solving problems.

• The symbol \( f \) is the name of the function, while \( x \) stands for the variable input or argument passed into this function.
• In the context of inverse functions, when we find an inverse, we write it as \( f^{-1}(x) \). This tells us that we are looking at a function that reverses \( f \).
• Using function notation, we can neatly and efficiently express operations and transformations, which helps in simplifying complex expressions.

Function notation is akin to a shorthand in mathematics. It helps in communicating ideas clearly, especially when sharing more detailed processes like finding inverses or describing transformations. When you see \( f(x) \) or \( f^{-1}(x) \), you know exactly the relationship being described without guessing.