/*! 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 1 The function \(f(x)=5^{x}\) is a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 1

The function \(f(x)=5^{x}\) is an exponential function with base _____; \(f(-2)=\) _____, \(f(0)=\) _____, \(f(2)=\) _____, and \(f(6)=\) _____.

Short Answer

Expert verified
The base is 5; \(f(-2) = \frac{1}{25}\), \(f(0) = 1\), \(f(2) = 25\), \(f(6) = 15625\).

Step by step solution

01

Identify the Base of the Exponential Function

The function given is \(f(x) = 5^x\). In an exponential function of the form \(a^x\), the base is the number \(a\). Hence, for \(f(x) = 5^x\), the base is 5.
02

Calculate \(f(-2)\)

Substitute \(-2\) for \(x\) in the function. So, \(f(-2) = 5^{-2}\). By the laws of exponents, \(5^{-2} = \frac{1}{5^2}\). Now calculate the power: \(5^2 = 25\). Thus, \(f(-2) = \frac{1}{25}\).
03

Calculate \(f(0)\)

Substitute 0 for \(x\) in the function. Thus, \(f(0) = 5^0\). According to the property of exponents, any non-zero number raised to the power of zero equals 1. Therefore, \(f(0) = 1\).
04

Calculate \(f(2)\)

Substitute 2 for \(x\) in the function. Thus, \(f(2) = 5^2\). Calculate the power: \(5^2 = 25\). Therefore, \(f(2) = 25\).
05

Calculate \(f(6)\)

Substitute 6 for \(x\) in the function. Thus, \(f(6) = 5^6\). Now, calculate the power: \(5^6 = 15625\). Hence, \(f(6) = 15625\).

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.

Understanding the Base of Exponential Functions
In an exponential function, such as the one in the exercise, the base is a crucial component. The function given is \(f(x) = 5^x\). Here, the "base" is the number that is repeatedly multiplied by itself, corresponding to the value of \(x\), known as the exponent. This base must always be greater than zero for the function to be defined.The base of an exponential function affects the growth or decay of the function:
  • If the base is greater than 1, the function represents exponential growth, meaning it will increase rapidly as \(x\) gets larger.
  • If the base is between 0 and 1, it represents exponential decay, where the function will decrease and approach zero as \(x\) increases.
For our function \(f(x) = 5^x\), the base is 5, indicating that with each increase in \(x\), the function will grow steeply.
Applying the Laws of Exponents
The laws of exponents provide a set of rules that simplify calculations involving exponential expressions. These rules are essential when working with functions like \(f(x) = 5^x\).Here are some key laws used in the solution:
  • Product of Powers: When multiplying like bases, you add the exponents, \(a^m \times a^n = a^{m+n}\).
  • Quotient of Powers: When dividing like bases, subtract the exponents, \(\frac{a^m}{a^n} = a^{m-n}\).
  • Power of a Power: When raising a power to another power, multiply the exponents, \((a^m)^n = a^{mn}\).
  • Negative Exponent: A negative exponent indicates the reciprocal of the positive exponent, \(a^{-n} = \frac{1}{a^n}\).
  • Zero Exponent: Any non-zero base raised to the power of zero equals 1, \(a^0 = 1\).
In the exercise, when you calculate \(f(-2) = 5^{-2}\), the negative exponent law tells us to take the reciprocal, resulting in \(\frac{1}{5^2}\). Similarly, for \(f(0) = 5^0\), the zero exponent law gives us a result of 1.
Steps to Calculate Exponents
Calculating exponents involves applying the base and exponent to derive a result. It's straightforward once you understand the process and rules.Here's a simple guide to calculating exponents as shown in the exercise:
  • Identify the base and the exponent. For example, in \(5^2\), 5 is the base and 2 is the exponent.
  • Multiply the base by itself as many times as the exponent indicates. So, \(5^2 = 5 \times 5\).
  • Simplify the expression to get the result. Here, \(5 \times 5 = 25\).
In the exercise, when calculating \(f(6) = 5^6\), you multiply 5 by itself six times, resulting in a large number, 15625. This method is critical to solving and understanding exponential functions efficiently. Practice these steps to become more comfortable with finding powers of numbers.

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

Difficulty of a Task The difficulty in "acquiring a target" such as using your mouse to click on an icon on your computer screen) depends on the distance to the target and the size of the target. According to Fits's Law, the index of difficulty (ID) is given by $$ \mathrm{ID}=\frac{\log (2 A / W)}{\log 2} $$ where \(W\) is the width of the target and \(A\) is the distance to the center of the target. Compare the difficulty of clicking on an icon that is 5 \(\mathrm{mm}\) wide to clicking on one that is 10 \(\mathrm{mm}\) wide. In each case, assume that the mouse is 100 \(\mathrm{mm}\) from the icon.

Population of a Country The population of a country has a relative growth rate of 3\(\%\) per year. The government is trying to reduce the growth rate to \(2 \% .\) The population in 1995 was approximately 110 million. Find the projected population for the year 2020 for the following conditions. (a) The relative growth rate remains at 3\(\%\) per year. (b) The relative growth rate is reduced to 2\(\%\) per year.

\(29-43\) . These exercises deal with logarithmic scales. Earthquake Magnitudes If one earthquake is 20 times as intense as another, how much larger is its magnitude on the Richter scale?

Compare the rates of growth of the functions \(f(x)=\ln x\) and \(g(x)=\sqrt{x}\) by drawing their graphs on a common screen using the viewing rectangle \([-1,30]\) by \([-1,6] .\)

A sum of \(\$ 1000\) was invested for 4 years, and the interest was compounded semiannually. If this sum amounted to \(\$ 1435.77\) in the given time, what was the interest rate?
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

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.