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Chapter 3: Problem 19

Let \(f(x)=2 x^{3}+1 .\) Find \(f^{-1}(x)\)

Short Answer

Expert verified
The inverse function is \(f^{-1}(x) = \sqrt[3]{\frac{x - 1}{2}}\).

Step by step solution

01

Understand the Problem

The goal is to find the inverse function of the given function \(f(x) = 2x^3 + 1\). The inverse function, \(f^{-1}(x)\), will essentially reverse the operation of \(f(x)\).
02

Replace \(f(x)\) with \(y\)

Set \(y = f(x) = 2x^3 + 1\). This allows us to solve for \(x\) in terms of \(y\), which is the first step in finding the inverse function.
03

Solve for \(x\)

Rearrange the equation \(y = 2x^3 + 1\) to solve for \(x\):1. Subtract 1 from both sides: \(y - 1 = 2x^3\).2. Divide by 2: \(\frac{y - 1}{2} = x^3\).3. Take the cube root of both sides: \(x = \sqrt[3]{\frac{y - 1}{2}}\).
04

Write the Inverse Function

Since we solved for \(x\) in terms of \(y\), we can now express \(f^{-1}(x)\) by replacing \(y\) with \(x\):\[ f^{-1}(x) = \sqrt[3]{\frac{x - 1}{2}} \]
05

Verify the Inverse Function

To verify, check if \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\):1. Let \(f^{-1}(x) = \sqrt[3]{\frac{x - 1}{2}}\). Substitute into \(f(x)\):\(f(f^{-1}(x)) = 2\left(\sqrt[3]{\frac{x - 1}{2}}\right)^3 + 1\).Simplify to get \(x\).2. Verify the reverse: Substitute \(f(x)\) back into \(f^{-1}\):\(f^{-1}(f(x)) = \sqrt[3]{\frac{(2x^3 + 1) - 1}{2}} = x\).Both checks confirm that \(f(x)\) and \(f^{-1}(x)\) are indeed inverses.

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Key Concepts

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

Function Composition
Function composition is a way of combining two functions to create a new function. The notation for function composition is usually written as \((f \circ g)(x)\), which can be read as "\(f\) of \(g\) of \(x\)." This means that you first apply the function \(g\) to the variable \(x\), and then apply the function \(f\) to the result.
  • If \(\ g(x) = x^2 \) and \(f(x) = x + 3\), then \((f \circ g)(x) = f(g(x)) = f(x^2) = x^2 + 3\).
  • Function composition is not commutative; \(f(g(x))\) is generally not the same as \(g(f(x))\).
Understanding function composition is crucial, especially when dealing with inverse functions. You use it to verify whether two functions are indeed inverses of each other, ensuring \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\). This property helps confirm that applying both functions in succession returns the original input value.
Cubic Functions
Cubic functions are polynomial functions of degree three, and they have the general form \(f(x) = ax^3 + bx^2 + cx + d\). These functions can model various real-world phenomena, such as projectile motion and population growth over time. In the exercise above, the function \(f(x) = 2x^3 + 1\) is an example of a cubic function.
  • Cubic functions can have one real root and two complex conjugate roots, or all three can be real.
  • They have a characteristic S-shape curve called a cubic curve, which can have at most two turning points.
By analyzing the symmetry and critical points of a cubic function, you can understand its behavior and find inverse functions. Since cubic functions have an odd degree, they are invertible as long as they are one-to-one. This is important for determining their inverse, such as the function \(f(x) = 2x^3 + 1\), which is inverted in this problem.
Verification of Inverse Functions
Verification of inverse functions checks if two functions nullify each other's effects when composed together. This is a crucial step to ensure the accuracy of an inverse function.
  • To verify that \(f(x)\) and \(f^{-1}(x)\) are inverses, compute \(f(f^{-1}(x))\) and \(f^{-1}(f(x))\).
  • Both expressions should simplify back to \(x\), confirming that one function exactly reverses the operation of the other.
In the exercise, the inverse function \(f^{-1}(x) = \sqrt[3]{\frac{x - 1}{2}}\) is subjected to such checks. By substituting into \(f(x)\) and \(f^{-1}(x)\), you demonstrate that the operations cancel each other out, as both calculations return the original value \(x\). This ensures that the determined inverse is indeed correct, and strengthens your understanding of how inverse functions work by testing their accuracy and validity through composition.

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Most popular questions from this chapter

Let \(T(x)=2 x^{2}-3 x .\) Find (and simplify) each expression. (a) \(T(x+2)\) (b) \(T(x-2)\) (c) \(T(x+2)-T(x-2)\)

Let \(f(x)=1 / x .\) Find a number \(b\) so that the average rate of change of \(f\) on the interval \([1, b]\) is \(-1 / 5\)

Use the given function and compute the first six iterates of each initial input \(x_{0}\). In cases in which a calculator answer contains four or more decimal places, round the final answer to three decimal places. (However, during the calculations, work with all of the decimal places that your calculator affords.) \(f(x)=2 x\) (a) \(x_{0}=1\) (b) \(x_{0}=0\) (c) \(x_{0}=-1\)

Use this definition: A prime number is a positive whole number with no factors other than itself and \(1 .\) For example, \(2,13,\) and 37 are primes, but 24 and 39 are not. \(B y\) convention 1 is not considered prime, so the list of the first few primes is as follows: \(2,3,5,7,11,13,17,19,23,29, \ldots\) Let \(f\) be the function that assigns to each natural number \(x\) the number of primes that are less than or equal to \(x .\) For example, \(f(12)=5\) because, as you can easily check, five primes are less than or equal to \(12 .\) Similarly, \(f(3)=2\) because two primes are less than or equal to \(3 .\) Find \(f(8)\) \(f(10),\) and \(f(50)\)

Assume that \((a, b)\) is a point on the graph of \(y=f(x),\) and specify the corresponding point on the graph of each equation. [For example, the point that corre- sponds to \((a, b)\) on the graph of \(y=f(x-1)\) is \((a+1, b).\)] (e) \(y=f(-x)\) (g) \(y=f(3-x)\) (f) \(y=-f(-x)\) (h) \(y=-f(3-x)+1\) (a) \(y=f(-x)+2\) (d) \(y=1-f(x+1)\) (b) \(y=-f(-x)+2\) (e) \(y=f(1-x)\) (c) \(y=-f(x-3)\) (f) \(y=-f(1-x)+1\)
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