/*! 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 72 Find the inverse function of \(f... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 72

Find the inverse function of \(f\). \(f(x)=3^{x+1}\)

Short Answer

Expert verified
The inverse function is \( f^{-1}(x) = \log_3(x) - 1 \).

Step by step solution

01

Identify the Function

We start with the function given: \[ f(x) = 3^{x+1} \]Our goal is to find the inverse of this function, denoted as \( f^{-1}(x) \). An inverse function essentially reverses the effect of the original function.
02

Replace f(x) with y

To indicate that the function holds an equality, we replace \( f(x) \) with \( y \):\[ y = 3^{x+1} \] This is a crucial step to solve for \( x \) in terms of \( y \), which will help us find \( f^{-1}(x) \).
03

Solve for x in terms of y

Now, we'll solve the equation \( y = 3^{x+1} \) for \( x \). First, take the logarithm base 3 of both sides to eliminate the exponent:\[ \log_3(y) = x + 1 \] Next, solve for \( x \) by isolating it on one side:\[ x = \log_3(y) - 1 \]
04

Express the Inverse Function

With \( x = \log_3(y) - 1 \), we now replace \( x \) with \( f^{-1}(x) \) and \( y \) with \( x \) to express the inverse function:\[ f^{-1}(x) = \log_3(x) - 1 \] This is the inverse function.
05

Verify the Inverse Function

To ensure that \( f^{-1}(x) \) is correct, check that applying \( f \) and then \( f^{-1} \) returns the original value:1. Calculate \( f(f^{-1}(x)) \): \[ f(f^{-1}(x)) = 3^{\log_3(x) - 1 + 1} = x \]2. Calculate \( f^{-1}(f(x)) \): \[ f^{-1}(3^{x+1}) = \log_3(3^{x+1}) - 1 = x + 1 - 1 = x \]Both checks show the function and its inverse correctly reverse each other's effect.

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.

Exponential Functions
Exponential functions are a type of mathematical function where the variable is in the exponent. In the function \( f(x) = 3^{x+1} \), the base is 3, and the exponent is \(x+1\). This means that as the variable \( x \) changes, the value of the function increases or decreases exponentially. There are a few key characteristics of exponential functions:
  • Growth Rate: Exponential functions grow by constant percentages, rather than constant differences. As \( x \) increases, \( f(x) \) grows faster compared to linear or quadratic functions.
  • Base Effects: The base (in this case, 3) affects the rate at which the function grows. Different bases result in different rates of growth or decay.
  • Horizontal Asymptote: As \( x \) approaches negative infinity, \( f(x) \) approaches zero, but never actually reaches it.
Understanding these characteristics is crucial for identifying and solving problems involving exponential functions, especially when dealing with their inverses.
Logarithms
Logarithms are the inverse of exponential functions. They take the form \( \log_b(a) \), which answers the question, "To what power must the base \( b \) be raised, to produce \( a \)?" In the context of our original function \( f(x) = 3^{x+1} \), taking the logarithm base 3 is a strategic move to solve for \( x \) when the function itself has \( x \) in its exponent.Logarithms are vital in unraveling exponential expressions:
  • Inverse Relationship: The logarithmic and exponential functions are inverse operations. Applying a logarithmic operation can cancel out an exponential function and vice versa.
  • Properties of Logarithms: There are several rules, such as \( \log_b(mn) = \log_b(m) + \log_b(n) \) and \( b^{\log_b(a)} = a \), which simplify calculations involving logarithms.
  • Solving Equations: By taking the logarithm of both sides, as shown in the solution, we turn complex exponential equations into manageable linear equations.
These features make logarithms particularly powerful tools in mathematics for functions, especially when determining the inverse.
Function Verification
Function verification is a vital step in ensuring the accuracy of an inverse function. When we find the inverse of a function, such as \( f(x) = 3^{x+1} \), it is essential to check that this inverse accurately reverses the original function.The process of function verification involves two main steps:
  • Substitution and Simplification: Calculate \( f(f^{-1}(x)) \). Substitute the inverse into the original function and simplify. For our inverse, \( f^{-1}(x) = \log_3(x) - 1 \), this is confirmed by showing that \( 3^{\log_3(x) - 1 + 1} = x \).
  • Reverse Substitution: Calculate \( f^{-1}(f(x)) \). Substitute the function into its inverse and simplify. For \( f(x) = 3^{x+1} \), this step involves showing \( \log_3(3^{x+1}) - 1 = x \).
Both of these calculations should result in the term \( x \), demonstrating that each function effectively undoes the other. Through these checks, we validate that the found inverse function is indeed correct and reliable.

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

Comparing Logarithms Which is larger, log, 17 or \(\log _{5} 24 ?\) Explain your reasoning.

\(25-28=\) These exercises use Newton's Law of Cooling. Cooling Turkey A roasted turkey is taken from an oven when its temperature has reached \(185^{\circ} \mathrm{F}\) and is placed on a table in a room where the temperature is \(75^{\circ} \mathrm{F} .\) (a) If the temperature of the turkey is \(150^{\circ} \mathrm{F}\) after half an hour, what is its temperature after 45 \(\mathrm{min}\) ? (b) When will the turkey cool to \(100^{\circ} \mathrm{F} ?\)

Atmospheric pressure \(P\) (in kilo-pascals, kPa) at altitude \(h\) (in kilometers, km) is governed by the formula $$\ln \left(\frac{P}{P_{0}}\right)=-\frac{h}{k}$$ where \(k=7\) and \(P_{0}=100 \mathrm{kPa}\) are constants. (a) Solve the equation for \(P .\) (b) Use part (a) to find the pressure \(P\) at an altitude of 4 \(\mathrm{km}\) .

Use a calculator to evaluate the expression, correct to four decimal places. $$ \begin{array}{llll}{\text { (a) } \log 50} & {\text { (b) } \log \sqrt{2}} & {\text { (c) } \log (3 \sqrt{2})}\end{array} $$

Use the definition of the logarithmic function to find \(x\). $$ \begin{array}{ll}{\text { (a) } \log _{4} 2=x} & {\text { (b) } \log _{4} x=2}\end{array} $$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Statistics

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.