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Chapter 1: Problem 14

For each function below, find \(f^{-1}(x)\). $$ f(x)=x+5 $$

Short Answer

Expert verified
The inverse function is \(f^{-1}(x) = x - 5\).

Step by step solution

01

Understand the Problem

We need to find the inverse function \(f^{-1}(x)\) for the given function \(f(x) = x + 5\). The inverse function essentially undoes what the original function does.
02

Set Up the Equation

To find the inverse, start by writing the equation \(f(x) = y = x + 5\). In terms of inverse functions, we swap \(x\) and \(y\) and solve for the new \(x\). Thus, rewrite the equation as \(x = y + 5\).
03

Solve for the Inverse Function

Now, solve the equation \(x = y + 5\) for \(y\) to find the inverse function. Subtract 5 from both sides to get \(y = x - 5\). Therefore, the inverse function is \(f^{-1}(x) = x - 5\).
04

Verify the Inverse Function

Check if the composition of \(f(x)\) and \(f^{-1}(x)\) yields the identity function. Compute \(f(f^{-1}(x)) = f(x - 5) = (x - 5) + 5 = x\) and verify \(f^{-1}(f(x)) = f^{-1}(x + 5) = (x + 5) - 5 = x\). Both check to the identity function \(x\).

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Key Concepts

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

Functions
A function is a special kind of relation between two sets. Specifically, it's a rule that assigns each element from a set, known as the domain, to exactly one element in another set, known as the range. Basic characteristics of functions:
  • A function has a specific input (domain) and output (range).
  • Every input must have a single output.
  • The notation \(f(x)\) is often used to represent a function.
In our exercise, the function given is \(f(x) = x + 5\). Here, for every input \(x\), the function outputs a value that is 5 more than \(x\). For example, if we input 3, the result from the function would be 8.Understanding how functions work allows us to manipulate them to find inverses, solve equations, and more. Functions map clear paths from one value to another, making them critical in mathematics.
Equations
Equations are foundational in mathematics; they assert that two expressions are equal, often used to find unknown values. In dealing with functions and their inverses, equations help us by establishing relationships:
  • In our example, to find the inverse function, we begin with the equation \(y = x + 5\).
  • By swapping \(x\) and \(y\), the equation becomes \(x = y + 5\).
  • We then solve for \(y\) by subtracting 5 from both sides, arriving at \(y = x - 5\).
This process transforms the initial equation into a new one, revealing the inverse function \(f^{-1}(x) = x - 5\). Equations allow us to unravel and step through relationships systematically. They are key in transitioning from the original function to its inverse.
Identity Function
The identity function is a special type of function that maps every element to itself. In simple terms, an identity function is expressed as \(f(x) = x\).When we find an inverse, we're interested in checking if applying a function and then its inverse brings us back to our original starting point, which is essentially what the identity function represents.
  • When we compose a function \(f(x)\) with its inverse \(f^{-1}(x)\), the result should be the identity function.
  • In our exercise, we verified this by demonstrating that \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\).
This verifies that our inverse \(f^{-1}(x) = x - 5\) correctly complements \(f(x) = x + 5\). The identity function serves as an important check to ensure we've found the correct inverse, reinforcing the symbiotic relationship between a function and its inverse.

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Most popular questions from this chapter

Find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\) $$ h(x)=3+\sqrt{x-2} $$

Sketch a graph of each piecewise function. $$ f(x)=\left\\{\begin{array}{lll} 4 & \text { if } & x<0 \\ \sqrt{x} & \text { if } & x \geq 0 \end{array}\right. $$

Describe how each function is a transformation of the original function \(f(x)\). $$ f(x)-2 $$

Find the domain of each function. $$ f(x)=5 \sqrt{x+3} $$

Describe how each function is a transformation of the original function \(f(x)\). $$ f(x+4)-1 $$
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