/*! 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 3 Answer each of the following. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 3

Answer each of the following. If \(f(x)=5^{x},\) what is the rule for \(f^{-1}(x) ?\)

Short Answer

Expert verified
The rule for \(f^{-1}(x)\) is \(\frac{\ln(x)}{\ln(5)}\).

Step by step solution

01

Understand the Given Function

The given function is exponential: \(f(x) = 5^x\). Understand that finding the inverse function requires determining \(y\) such that \(y = 5^x\), then solving for \(x\) in terms of \(y\).
02

Substitute and Solve

Rewrite the function with \(y\) instead of \(f(x)\). Therefore, \(y = 5^x\). The goal is to solve for \(x\) as a function of \(y\).
03

Apply the Logarithm

To isolate \(x\), take the natural logarithm (log base \(e\)) of both sides: \(\ln(y) = \ln(5^x)\).
04

Use Logarithm Properties

\(\ln(5^x)\) can be rewritten using the power rule for logarithms: \(\ln(5^x) = x \ln(5)\). Thus, \(\ln(y) = x \ln(5)\).
05

Isolate \(x\)

Solve for \(x\) by dividing both sides by \(\ln(5)\): \(x = \frac{\ln(y)}{\ln(5)}\).
06

Write the Inverse Function

The inverse function \(f^{-1}(x)\) is obtained by substituting \(y\) back with \(x\): \(f^{-1}(x) = \frac{\ln(x)}{\ln(5)}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Inverse Functions
Inverse functions essentially reverse the effect of the original function. If you have a function that transforms input into output, an inverse function takes that output back to the original input. For example, if a function says that when you put in 2, you get 4, the inverse function would say that when you put in 4, you get back 2.
To find an inverse function, you swap the roles of input and output and solve for the new output. This can be particularly useful in many scientific and engineering fields where reverse calculations are often needed.
Exponential Functions
An exponential function is a type of function where the variable appears as an exponent. For instance, in the function given as the exercise, we have: \(f(x) = 5^x\). Here, 5 is the base, and x is the exponent.
Exponential functions grow very quickly, which makes them useful for modeling processes that involve rapid growth or decay, such as population growth or radioactive decay.
Key characteristics of exponential functions include:
  • Base greater than 1: Shows growth
  • Base between 0 and 1: Shows decay
  • Always positive for real number inputs
Understanding how to manipulate these functions is critical for solving more complex mathematical problems.
Natural Logarithms and Logarithm Properties
Natural logarithms (logarithms with base e) are inverse operations to exponential functions. The natural logarithm is denoted as \(\ln(x)\). In the exercise, we used this to find the inverse of an exponential function.
The logarithm properties are essential tools for simplifying and solving logarithmic and exponential equations:
  • Product Property: \(\ln(ab) = \ln(a) + \ln(b)\)
  • Quotient Property: \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\)
  • Power Property: \(\ln(a^b) = b \ln(a)\)
Using these properties, you can break down and solve complex equations. For example, in our exercise, we used the Power Property to re-write \(\ln(5^x) = x \ln(5)\), which then allowed solving for x. Understanding these properties can help simplify and solve various mathematical problems efficiently.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Answer each of the following. For a one-to-one function \(f,\) find \(\left(f^{-1} \circ f\right)(2),\) where \(f(2)=3\).

For each function as defined that is one-to-one, (a) write an equation for the inverse function in the form \(y=f^{-1}(x),\) (b) graph \(f\) and \(f^{-1}\) on the same axes, and \((c)\) give the domain and the range of \(f\) and \(f^{-1}\). If the function is not one-to-one, say so. $$f(x)=-4 x+3$$

At the World Championship races held at Rome's Olympic Stadium in \(1987,\) American sprinter Carl Lewis ran the 100 -m race in 9.86 sec. His speed in meters per second after \(t\) seconds is closely modeled by the function $$f(t)=11.65\left(1-e^{-t / 1.27}\right)$$ (Source: Banks, Robert B., Towing Icebergs, Falling Dominoes, and Other Adventures in Applied Mathematics, Princeton University Press.) (a) How fast was he running as he crossed the finish line? (b) After how many seconds was he running at the rate of \(10 \mathrm{m}\) per sec?

Use another type of logistic function. Tree Growth The height of a certain tree in feet after \(x\) years is modeled by $$ f(x)=\frac{50}{1+47.5 e^{-0.22 x}} $$ (a) Make a table for \(f\) starting at \(x=10,\) and incrementing by \(10 .\) What appears to be the maximum height of the tree? (b) Graph \(f\) and identify the horizontal asymptote. Explain its significance. (c) After how long was the tree 30 ft tall?

Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. $$\log _{0.32} 5$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.