/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 52 A one-to-one function is given. ... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 52

A one-to-one function is given. Write an equation for the inverse function. \(n(x)=2 x^{3}-5\)

Short Answer

Expert verified
n^{-1}(x) = \(\sqrt[3]{\frac{x+5}{2}}\)

Step by step solution

01

Understand the given function

The given function is (x) = 2x^3 - 5. To find the inverse function, we need to switch the roles of x and n(x).
02

Replace n(x) with y

Rewrite the function as y = 2x^3 - 5.
03

Swap y and x

Interchange the variables x and y to get x = 2y^3 - 5.
04

Solve for y

Isolate y by first adding 5 to both sides: x + 5 = 2y^3. Then, divide both sides by 2: (x + 5)/2 = y^3. Finally, take the cube root of both sides: y = \(\sqrt[3]{\frac{x+5}{2}}\).
05

Write the inverse function

Replace y with n^{-1}(x) to get the inverse function: n^{-1}(x) = \(\sqrt[3]{\frac{x+5}{2}}\).

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Key Concepts

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

One-to-One Function
A one-to-one function is a type of function where each element in the domain is paired with a unique element in the range. This means that no two different inputs produce the same output. An easy way to visualize this is by considering a horizontal line test. If a horizontal line crosses the graph of the function more than once, the function is not one-to-one. This property is crucial for finding an inverse because only one-to-one functions have inverses.
Finding Inverse
Finding the inverse of a function involves a series of steps to essentially reverse the operation of the original function. The process includes interchanging the dependent and independent variables and then solving for the new dependent variable. In the given problem, the function is: (x) = 2x^3 - 5To find its inverse, we start by setting the function equal to y: y = 2x^3 - 5Next, we swap x and y: x = 2y^3 - 5Finally, we solve for y, which involves several algebraic manipulations. This new equation with y isolated represents the inverse function.鈥� }, {
Function Notation
Function notation is a way to express functions in a clear and concise manner. In the equation given (x) = 2x^3 - 5, (x) is the function notation where 'n' denotes the function name and 'x' represents the input variable. When finding an inverse function, the notation used is typically ^{-1}(x), where the '^{-1}' indicates the inverse. For instance, in this problem, the inverse function is written as ^{-1}(x) = ( x+5 /2)鈥�. Function notation helps to organize and simplify working with complex functions and their inverses through clear and understandable symbols.
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying equations to solve for a specific variable. This is a critical skill when finding the inverse of a function. In the problem, after swapping x and y, we get:x = 2y^3 - 5We need to solve this for y. First, add 5 to both sides to isolate the term with y:x + 5 = 2y^3Next, divide both sides by 2:(x + 5) / 2 = y^3Finally, take the cube root of both sides to solve for y:y = {x+5 /2}of algebraic manipulation helps to isolate y, giving us the inverse function.

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Most popular questions from this chapter

Show that \(2\left(\frac{e^{x}-e^{-x}}{2}\right)\left(\frac{e^{x}+e^{-x}}{2}\right)=\frac{e^{2 x}-e^{-2 x}}{2}\).

The half-life of radium-226 is 1620 yr. Given a sample of \(1 \mathrm{~g}\) of radium \(-226,\) the quantity left \(Q(t)\) (in g) after \(t\) years is given by \(Q(t)=\left(\frac{1}{2}\right)^{t / 1620}\) a. Convert this to an exponential function using base \(e\). b. Verify that the original function and the result from (a) yield the same result for \(Q(0), Q(1620),\) and part \(Q(3240) .\) (Note: There may be round-off error.)

Given the functions defined by \(f(x)=2 x-1\) and \(g(x)=\frac{x+1}{2}\), a. Graph \(y=f(x), y=g(x),\) and the line \(y=x .\) Does the graph suggest that \(f\) and \(g\) are inverses? Why? b. Enter the following functions into the graphing editor. ( $$\mathrm{Y}_{1}=2 x-1$$ \(\mathrm{Y}_{2}=(x+1) / 2\) \(\mathrm{Y}_{3}=\mathrm{Y}_{1}\left(\mathrm{Y}_{2}\right)\) \(\mathrm{Y}_{4}=\mathrm{Y}_{2}\left(\mathrm{Y}_{1}\right)\) c. Create a table of points showing \(Y_{3}\) and \(Y_{4}\) for several values of \(x\). (Hint: Use the right and left arrows to scroll through the table editor to show functions \(Y_{3}\) and \(Y_{4}\).) Does the table suggest that \(f\) and \(g\) are inverses? Why?

A table of data is given. a. Graph the points and from visual inspection, select the model that would best fit the data. Choose from $$\begin{array}{ll} y=m x+b \text { (linear) } & y=a b^{x} \text { (exponential) } \\ y=a+b \ln x \text { (logarithmic) } & y=\frac{c}{1+a e^{-b x}} \text { (logistic) } \end{array}$$ b. Use a graphing utility to find a function that fits the data. $$ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 640 \\ \hline 20 & 530 \\ \hline 40 & 430 \\ \hline 50 & 360 \\ \hline 80 & 210 \\ \hline 100 & 90 \\ \hline \end{array} $$

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(10^{5+8 x}+4200=84,000\)
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