/*! 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 49 Find the inverse of each functio... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 49

Find the inverse of each function. $$p(x)=\sqrt[4]{x}$$

Short Answer

Expert verified
The inverse function is \(p^{-1}(x) = x^4\).

Step by step solution

01

Understand the Function

The given function is \(p(x) = \sqrt[4]{x}\). This function takes the fourth root of an input value \(x\).
02

Replace Function Notation with y

Rewrite the function using \(y\) instead of \(p(x)\). So, \(y = \sqrt[4]{x}\).
03

Swap x and y

To find the inverse, swap \(x\) and \(y\). The equation becomes \(x = \sqrt[4]{y}\).
04

Solve for y

Raise both sides of the equation to the fourth power to solve for \(y\). So, \(x^4 = y\).
05

Write the Inverse Function

Replace \(y\) with the inverse function notation. Thus, the inverse function is \(p^{-1}(x) = x^4\).

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Key Concepts

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

Function Notation
Function notation is a way to describe a function using symbols. For instance, in our problem, we have the function notation of the form \( p(x) \). This means that \( p \) is a function with \( x \) as the input variable. With function notation, we can clearly represent the operations performed on the input.

  • For example, \( p(x) = \sqrt[4]{x}\) tells us that our function takes the fourth root of \( x \).
  • Switching between expressions like \( y = p(x) \) helps make equations more manageable.
  • Swapping symbols like \( x \) and \( y \) is crucial when finding inverse functions.

It鈥檚 helpful to get comfortable with function notation as it鈥檚 widely used in algebra and calculus. Once you know this, you鈥檒l find it easier to manipulate and understand functions.
Solving Equations
Solving equations is about finding the values of variables that make the equation true. In our provided solution, we need to solve for \( y \) after swapping \( x \) and \( y \).

  • Start with \( x = \sqrt[4]{y}\) after swapping.
  • To isolate \( y \), we raise both sides to the fourth power: \( x^4 = y \).

This process, often called 'isolating the variable', lets us express \( y \) in terms of \( x \). Make sure each step is clear and all operations follow algebraic principles. This ensures we find the correct inverse function.
Algebraic Manipulation
Algebraic manipulation involves rearranging equations to make them easier to solve. In our exercise on inverse functions, it鈥檚 crucial to manipulate equations correctly.

  • First, understand the given function: \( p(x) = \sqrt[4]{x} \).
  • Convert it by replacing \( p(x) \) with \( y \): \( y = \sqrt[4]{x} \).
  • Swap \( x \) and \( y \) to set the stage for finding the inverse: \( x = \sqrt[4]{y} \).
  • To solve for \( y \), raise both sides to the fourth power: \( x^4 = y \).
  • Finally, rewrite \( y \) as the inverse notation: \( p^{-1}(x) = x^4 \).

These steps showcase basic algebraic manipulation skills necessary for working with functions and their inverses. Practice helps in recognizing how and when to apply these manipulation techniques.

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Most popular questions from this chapter

Classify each function as either a linear, constant, quadratic, square-root, or absolute value function. $$f(x)=4$$

Find the requested values. See Example 4. If \(D\) varies directly with \(t\) and inversely with the square of \(s,\) and \(D=12.35\) when \(t=2.8\) and \(s=2.48,\) find \(D\) when \(t=5.63\) and \(s=6.81\).

Find \(f^{-1}\). Check that \(\left(f \circ f^{-1}\right)(x)=x\) and \(\left(f^{-1} \circ f\right)(x)=x\) Strategy for Finding \(f^{-1}\) by Switch-and Solve. $$f(x)=\frac{3}{x-4}$$

Find the inverse of each function. $$f(x)=2 \sqrt[3]{x}-1$$

Graph \(f(x)=|x|\) and \(g(x)=|x-20|+30\) on the same screen of your calculator. What transformations will transform the graph of \(f\) into the graph of \(g ?\)
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