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Chapter 3: Problem 16

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

Short Answer

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

Step by step solution

01

Replace f(x) with y

We replace the given function $$f(x)$$ with $$y$$: $$y=5+\sqrt{3x-2}$$
02

Swap x and y

Now, we swap $$x$$ and $$y$$ in the equation: $$x=5+\sqrt{3y-2}$$
03

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 $$3$$: $$3y=x^2-10x+27$$ $$y=\frac{1}{3}(x^2-10x+27)$$
04

Write the inverse function

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

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Key Concepts

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

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鈥檚 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鈥檛 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鈥檒l 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鈥檚 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.

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Most popular questions from this chapter

Find the approximate intervals on which the function is increasing, those on which it is decreasing, and those on which it is constant. $$f(x)=\frac{1}{x}$$

Determine the domain of the function according to the usual convention. $$f(x)=\sqrt{-x}+\frac{2}{x+1}$$

Find the approximate intervals on which the function is increasing, those on which it is decreasing, and those on which it is constant. $$f(x)=\sqrt{x}$$

Sketch the graph of a function \(f\) that satisfies these five conditions: (i) \(f(-1)=2\) (ii) \(f(x) \geq 2\) when \(x\) is in the interval \(\left(-1, \frac{1}{2}\right)\) (iii) \(f(x)\) starts decreasing when \(x=1\) (iv) \(f(3)=3=f(0)\) (v) \(f(x)\) starts increasing when \(x=5\) [Note: The function whose graph you sketch need not be given by an algebraic formula.]

In each part, compute \(f(a), f(b),\) and \(f(a+b),\) and determine whether the statement "f \((a+b)=f(a)+f(b)\) " is true or false for the given function. (a) \(f(x)=x^{2}\) (b) \(f(x)=3 x\) (c) \(f(x)=5\)
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