/*! 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 Choose the correct response in E... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 3

Choose the correct response in Exercises \(1-3\). The asymptote of the graph of \(f(x)=a^{x}\) A. is the \(x\) -axis B. is the \(y\) -axis C. has equation \(x=1\) D. has equation \(y=1\)

Short Answer

Expert verified
The asymptote of the graph of \(f(x) = a^x\) is the \(x\)-axis (Option A).

Step by step solution

01

- Understand Asymptotes

An asymptote of a graph is a line that the graph approaches but never touches. For exponential functions, like the given function, it's important to identify the behavior of the graph as it extends towards positive and negative infinity.
02

- Identify the Given Function

The given function is an exponential function: \[ f(x) = a^x \] where \(a\) is a positive constant.
03

- Analyze the Behavior of the Function

For large positive values of \(x\), the function \(f(x) = a^x\) grows rapidly. For large negative values of \(x\), the function \(f(x)\) approaches zero but never actually reaches zero.
04

- Determine the Asymptote

Since \(f(x)\) approaches zero as \(x\) goes to negative infinity, the horizontal asymptote is the \(x\)-axis, or the line \(y = 0\).
05

- Choose the Correct Response

Based on the analysis, the correct response for the asymptote of the function \(f(x) = a^x\) is option A, which states that the asymptote is the \(x\)-axis.

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
An exponential function is a mathematical expression in which a constant base is raised to a variable exponent. The general form of an exponential function is \[ f(x) = a^x \] where 'a' is a positive constant greater than 1 (if 'a' is less than 1, the function represents exponential decay).
Exponential functions grow or decay at rates proportional to their current value. They model a wide range of real-world phenomena, including population growth, radioactive decay, and interest compounding in finance.
Understanding exponential functions is crucial because they exhibit rapid changes: they expand quickly for positive values of 'x' and diminish swiftly as 'x' turns negative.
graph behavior
The behavior of the graph of an exponential function \(f(x) = a^x\) reveals essential insights.
For large positive values of 'x', the function value increases rapidly. This means that, graphically, the curve rises steeply as you move to the right.
Conversely, for large negative values of 'x', the function value gets closer and closer to zero without ever reaching it. Graphically, this results in the curve approaching the x-axis but never touching or crossing it. This behavior highlights a key feature of exponential functions: they exhibit asymptotic behavior, meaning they get infinitely close to a line but do not intersect it.
horizontal asymptote
A horizontal asymptote of a graph is a horizontal line that the graph approaches as the input (or x-values) either increase or decrease without bound.
In an exponential function \(f(x) = a^x\), for any positive value of 'a', as 'x' approaches negative infinity, the function values approach zero. This means that the horizontal asymptote is the x-axis or the line \(y = 0\).
Horizontal asymptotes are pivotal in understanding the end behavior of functions. They give us a notion of how the function behaves far out on the x-axis, simplifying analysis and predictions.
function analysis
Function analysis involves examining and breaking down the properties of a given function. For the exponential function \(f(x) = a^x\), key properties to consider include:
  • Identifying the base 'a', which influences the growth or decay rate.
  • Analyzing the behavior of the function as 'x' approaches positive and negative infinity.
  • Determining the horizontal asymptote, which, in this case, is the x-axis (or y = 0).
Combining these analyses helps in understanding the overall shape and behavior of the graph. By identifying critical points and asymptotes, we can gain a comprehensive understanding of the function鈥檚 behavior, predict its trends, and apply this knowledge to real-world situations.

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

To four decimal places, the values of \(\log _{10} 2\) and \(\log _{10} 9\) are $$ \log _{10} 2=0.3010 \quad \text { and } \quad \log _{10} 9=0.9542 $$ Evaluate each logarithm by applying the appropriate rule or rules from this section. $$ \log _{10} \sqrt[5]{2} $$

Consumers can now enjoy movies at home in elaborate home-theater systems. Find the average decibel level \(D=10 \log \left(\frac{I}{I_{0}}\right)\) for each movie with the given intensity \(I\) (a) Avatar; \(5.012 \times 10^{10} \mathrm{I}_{0}\) (b) Iron Man \(2 ; \quad 10^{10} I_{0}\) (c) Clash of the Titans; \(6,310,000,000 \mathrm{I}_{0}\)

Use the properties of logarithms to write each expression as a single logarithm. Assume that all variables are defined in such a way that the variable expressions are positive, and bases are positive numbers not equal to 1. $$ 3 \log _{a} 5-\frac{1}{2} \log _{a} 9 $$

To solve the equation \(5^{x}=7,\) we must find the exponent to which 5 must be raised in order to obtain \(7 .\) This is \(\log _{5} 7\) (a) Use the change-of-base rule and your calculator to find \(\log _{5} 7\) (b) Raise 5 to the number you found in part (a). What is your result? (c) Using as many decimal places as your calculator gives, write the solution set of \(\left.5^{x}=7 .\)

Solve equation. \(\log _{1 / 2} 8=x\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Pure 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.