/*! 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 231 For the following exercises, eva... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 231

For the following exercises, evaluate the given exponential functions as indicated, accurate to two significant digits after the decimal. $$f(x)=10^{x} \text { a. } x=-2 \text { b. } x=4 \text { c. } x=\frac{5}{3}$$

Short Answer

Expert verified
\( f(-2) = 0.01 \), \( f(4) = 10000 \), \( f\left(\frac{5}{3}\right) \approx 21.54 \).

Step by step solution

01

Understanding the Function

The function given is an exponential function, \( f(x) = 10^x \). This means for each input \( x \), we calculate \( 10 \) raised to the power of \( x \).
02

Calculating for \( x = -2 \)

To evaluate \( f(x) \) at \( x = -2 \), substitute \( x \) with \( -2 \) in the function. Therefore, \( f(-2) = 10^{-2} = \frac{1}{10^2} = \frac{1}{100} = 0.01 \).
03

Calculating for \( x = 4 \)

To evaluate \( f(x) \) at \( x = 4 \), substitute \( x \) with \( 4 \) in the function. Therefore, \( f(4) = 10^4 = 10000 \).
04

Calculating for \( x = \frac{5}{3} \)

To evaluate \( f(x) \) at \( x = \frac{5}{3} \), substitute \( x \) with \( \frac{5}{3} \) in the function. Therefore, use a calculator to find \( 10^{\frac{5}{3}} \approx 21.54 \).

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.

Evaluating Exponential Functions
Exponential functions are a key type of mathematical functions with the form \( f(x) = a^x \), where \( a \) is a positive constant, often referred to as the base, and \( x \) is the exponent. Evaluating exponential functions involves calculating the power of the base \( a \) raised to different values of \( x \).
To evaluate an exponential function:
  • Identify the base, which is the constant number that is repeatedly multiplied.
  • Determine the exponent, which dictates how many times the base is multiplied by itself.
  • Substitute the specific value into the exponent and carry out the calculation.
When evaluating, sometimes you'll encounter negative exponents, which indicate a reciprocal. For example, \( 10^{-2} = \frac{1}{10^2} \). For fractional exponents like \( \frac{5}{3} \), these can be evaluated using a calculator to approximate the value.
Exponential Growth and Decay
Exponential growth and decay describe processes that increase or decrease proportionally to the amount present. In the context of exponential functions, growth occurs when the base \( a \) is greater than 1, leading to an increasing trend, while decay happens for bases between 0 and 1, resulting in a decreasing trend.
Understanding these concepts:
  • Exponential growth can be illustrated by functions like \( 10^x \), where increasing \( x \) results in rapidly increasing values, as seen in \( 10^4 = 10000 \).
  • Exponential decay is observed in functions where the base is the reciprocal of some number greater than 1, such as \( 10^{-x} \), demonstrating a decrease as \( x \) increases (e.g., \( 10^{-2} = 0.01 \)).
These models are applied in various real-life scenarios, like population growth, radioactive decay, and interest computations.
Exponents and Powers
Exponents and powers are fundamental components of exponential functions and play a crucial role in mathematics. An exponent indicates how many times a base is used in multiplication. In the expression \( b^n \), \( b \) is the base, and \( n \) is the exponent, implying \( b \) is multiplied by itself \( n \) times.
Key aspects to remember:
  • Positive exponents indicate normal multiplication: \( 10^3 = 10 \times 10 \times 10 = 1000 \).
  • Negative exponents suggest division or reciprocals: \( 10^{-2} = \frac{1}{10^2} = \frac{1}{100} \).
  • Fractional exponents correspond to roots and powers: \( 10^{\frac{5}{3}} \) suggests taking the cube root of \( 10^5 \).
Mastering exponents and powers enables understanding and evaluation of complex exponential functions and lays the groundwork for advanced mathematical concepts.

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

Suppose that \(T=50+10 \sin \left[\frac{\pi}{12}(t-8)\right]\) is a mathematical model of the temperature (in degrees Fahrenheit) at \(t\) hours after midnight on a certain day of the week. a. Determine the amplitude and period. b. Find the temperature 7 hours after midnight. c. At what time does \(T=60^{\circ} ?\) d. Sketch the graph of \(T\) over \(0 \leq t \leq 24\)

[T] The demand \(D\) (in millions of barrels) for oil in an oil-rich country is given by the function \(D(p)=150 \cdot(2.7)^{-0.25 p}\) where \(p\) is the price (in dollars) of a barrel of oil. Find the amount of oil demanded (to the nearest million barrels) when the price is between \(\$ 15\) and \(\$ 20 .\)

The function \(H(t)=8 \sin \left(\frac{\pi}{6} t\right)\) models the height \(H(\text { in feet })\) of the tide \(t\) hours after midnight. Assume that \(t=0\) is midnight. a. Find the amplitude and period. b. Graph the function over one period. c. What is the height of the tide at 4:30 a.m.?

For the following problems, consider the population of Ocean City, New Jersey, which is cyclical by season. In reality, the overall population is most likely increasing or decreasing throughout each year. Let's reformulate the model as \(P(t)=82.5-67.5 \cos [(\pi / 6) t]+t, \quad\) where \(t\) is time in months \((t=0 \text { represents January } 1)\) and \(P\) is population (in thousands). When is the first time the population reaches \(200,000 ?\)

For the following exercises, use properties of logarithms to write the expressions as a sum, difference, and/or product of logarithms. $$ \log _{5} \sqrt{125 x y^{3}} $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Decision Maths

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.