Book cover for Algebra and Trigonometry Real Mathematics, Real People

Algebra and Trigonometry Real Mathematics, Real People

Ron Larson

ISBN #9781305071735

7th Edition

6,909 Questions

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234,597 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces exponential functions as a distinct type of transcendental function characterized by the variable in the exponent. Key properties include the function’s domain of (-?, ?), range of (0, ?), and the presence of a horizontal asymptote at y = 0. Essential techniques involve evaluation using calculators, graphing basics, and understanding transformations. Real-world applications such as modeling compound interest, population growth, and radioactive decay further demonstrate the practical significance of exponential functions, particularly through the natural exponential function f(x) = e^x.

Learning Objectives

1

Describe the definition and key properties of exponential functions, including the restrictions on the base.

2

Evaluate exponential functions for various types of exponents (including irrational and negative exponents) using a calculator.

3

Graph exponential functions and recognize their transformations such as shifts and reflections.

4

Analyze real-world applications of exponential functions in areas such as compound interest, population growth, and radioactive decay.

Key Concepts

CONCEPT

DEFINITION

Exponential Function

A function of the form f(x) = a^x, where a > 0 and a ≠ 1, with x as any real number.

Natural Exponential Function

The function f(x) = e^x, where e ≈ 2.71828; it is especially useful for modeling continuous growth or decay.

Transcendental Function

Functions that are not algebraic, such as exponential and logarithmic functions.

Graph Transformation

Changes to the parent graph (such as shifts, reflections, and stretches) resulting from modifications to the function's equation.

Compound Interest

A financial application where interest is added to the principal and then earns interest itself, modeled by A = P(1 + r/n)^(nt) for n compounding periods or A = P e^(rt) for continuous compounding.

Example Problems

Example 1

Fill in the blank(s). Exponential and logarithmic functions are examples of nonalgebraic functions, also called _________ functions.

Example 2

The exponential function $f(x)=e^{x}$ is called the _____________ function, and the base $e$ is called the ___________ base.

Example 3

What type of transformation of the graph of $f(x)=5^{x}$ is the graph of $f(x+1) ?$

Example 4

The formula $A=P e^{\pi}$ gives the balance $A$ of an account earning what type of interest?

Example 5

Use a calculator to evaluate the function at the indicated value of $x .$ Round your result to three decimal places. Value $x=6.8$ $x=\frac{1}{3}$ $x=-\pi$ $x=-\sqrt{2}$ Function $f(x)=3.4^{x}$
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Step-by-Step Explanations

QUESTION

Evaluate f(x) = 2^x at x = -3.1 using a calculator.

STEP-BY-STEP ANSWER:

Step 1: Identify the function f(x) = 2^x and substitute the value x = -3.1, getting f(-3.1) = 2^(-3.1).
Step 2: Ensure you input the negative exponent correctly—enclose the exponent in parentheses if necessary, e.g., 2^(-3.1).
Step 3: Use a calculator to compute 2^(-3.1) and note the approximate value, which is roughly 0.11663.
Final Answer: f(-3.1) ≈ 0.11663.

Evaluating an Exponential Function

QUESTION

Find the balance A after 5 years if a principal of $9000 is invested at an annual interest rate of 2.5% compounded annually.

STEP-BY-STEP ANSWER:

Step 1: Write the compound interest formula: A = P(1 + r/n)^(n*t). For annual compounding, n = 1.
Step 2: Substitute P = 9000, r = 0.025, n = 1, and t = 5 into the formula to obtain A = 9000(1.025)^5.
Step 3: Use a calculator to compute (1.025)^5 and multiply by 9000.
Final Answer: A ≈ $10,182.67.

Compound Interest Calculation

QUESTION

Describe how to obtain the graph of g(x) = f(x + 1) when given f(x) = 3^x.

STEP-BY-STEP ANSWER:

Step 1: Recognize that g(x) = f(x + 1) represents a horizontal shift.
Step 2: Since the transformation uses x + 1, this shifts the graph of f(x) one unit to the left.
Step 3: Plot several points from f(x) = 3^x and then shift them one unit left to obtain g(x).
Final Answer: The graph of g(x) is the graph of f(x) shifted one unit to the left.

Graph Transformation

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Common Mistakes

  • Confusing a constant function (like 1^x) with an exponential function.
  • Forgetting to include parentheses when entering negative or fractional exponents into a calculator, leading to order of operations errors.
  • Misunderstanding that a negative exponent does not yield a negative output but rather the reciprocal of the positive exponent.
  • Overlooking the impact of graph transformations, such as misidentifying shifts or reflections.
  • Assuming that greater compounding frequency leads to unlimited growth without recognizing the model’s asymptotic nature.