/*! 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 2 Exponential functions have a con... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 2

Exponential functions have a constant base and a variable __________

Short Answer

Expert verified
The variable in an exponential function is the exponent.

Step by step solution

01

Understanding the Concept of Exponential Functions

An exponential function is a mathematical expression in the form \( f(x) = a^x \), where \( a \) is a constant base greater than zero, and \( x \) is the exponent or power that can vary. The variable part is often the power, which determines the steepness and direction of the graph of the function.
02

Identifying the Variable

In the exponential function \( f(x) = a^x \), the base \( a \) is constant, and \( x \) is the exponent that varies. This means that for any such function, the variable is the exponent or power part \( x \).
03

Conclusion

Based on the definition and structure of exponential functions, the variable in an exponential function is the exponent.

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.

Constant Base
In exponential functions, the base remains constant throughout the expression. This constant can be any positive real number but is generally greater than zero.
For instance, in the function
  • \( f(x) = 2^x \), the base is 2,
  • in \( f(x) = 5^x \), the base is 5.
The constant base plays a crucial role as it influences the rate at which the function grows or decays.
A larger base results in faster growth or decay. This is because each increase in the exponent multiplies the result by the constant base value.
Understanding the constant base helps us predict the behavior of exponential functions effectively.
Variable Exponent
The defining characteristic of exponential functions is the variable in the exponent position. Unlike the base, the exponent changes, which is why it is called a variable exponent.
In the exponential function form \( f(x) = a^x \), the exponent \( x \) determines the value of the function.
This means the graph of an exponential function can change dramatically with small shifts in the exponent.
  • As \( x \) increases, the function exhibits growth if the base is greater than one.
  • Conversely, for a base between 0 and 1, the function demonstrates decay as \( x \) increases.
The variable exponent is the key to the dynamic and variable nature of exponential functions, allowing for diverse applications ranging from population models to radioactive decay scenarios.
Graph of the Function
The graph of an exponential function provides a visual representation of how the function behaves as the variable exponent changes.
When graphing an exponential function like \( f(x) = a^x \), several important features stand out:
  • The graph passes through the point (0,1), since \( a^0 = 1 \) for any non-zero base \( a \).
  • If the base \( a > 1 \), the graph shows exponential growth, curving upwards as \( x \) increases.
  • If the base \( 0 < a < 1 \), the graph demonstrates exponential decay, curving downwards towards the x-axis as \( x \) increases.
The general shape of the graph helps understand how exponential functions either increase or decrease rapidly.
Additionally, exponential graphs never touch the x-axis, which means they have a horizontal asymptote along the x-axis. This feature highlights the function's tendency to approach but never reach zero, embodying the concept of infinite growth or decay.

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

Find \(f(x)\) and \(g(x)\) such that \(h(x)=(f \circ g)(x) .\) Answers may vary. See Example 5. $$ h(x)=x^{5}+9 $$

Epidemics. The spread of hoof-and-mouth disease through a herd of cattle can be modeled by the function \(P(t)=2 e^{0.27 t}\) ( \(t\) is in days). If a rancher does not quickly treat the two cows that now have the disease, how many cattle will have the disease in 12 days?

Cross Country Skiing. The function \(H(s)=-47.73+107.38\) ln \(s\) approximates the heart rate (in beats/minute) for an Olympic-class cross country skier traveling at \(s\) miles per hour, where \(s>5\) mph. Find the heart rate of a skier traveling at a rate of 7.5 miles per hour.

Let \(f(x)=\frac{1}{x}\) and \(g(x)=\frac{1}{x^{2}} .\) Find each of the following. $$ (g \circ f)\left(\frac{1}{3}\right) $$

Let \(f(x)=3 x-2\) and \(g(x)=x^{2}+x .\) Find each of the following. $$ (g \circ f)(4) $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Applied Mathematics

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.