Problem 14 Graph the function, not by plott... [FREE SOLUTION]

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Precalculus Mathematics for Calculus

James Stewart, Lothar Redlin, Saleem Watson

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 4: Problem 14

Graph the function, not by plotting points, but by starting from the graph of $y=e^{x}$ in Figure $1 .$ State the domain, range, and asymptote. $$g(x)=-e^{x-1}-2$$

Short Answer

Expert verified

Domain: all real numbers, Range: $y < -2$, Asymptote: $y = -2$.

Step by step solution

Identify the Base Graph

The base graph is given as $ y = e^x $. This is the standard exponential function with a horizontal asymptote at $ y = 0 $, a domain of all real numbers ($ x \in \mathbb{R} $), and a range of positive real numbers ($ y > 0 $).

Apply Horizontal Transformation

The given function is $ g(x) = -e^{x-1} - 2 $. Notice the $ x-1 $ part inside the exponent. This indicates a horizontal shift to the right by 1 unit. This doesn't affect the domain or the asymptote, but it shifts the graph.

Apply Reflection and Vertical Shift

The negative sign before $ e^{x-1} $ indicates a reflection about the x-axis. The $-2$ indicates a vertical shift downward by 2 units. The horizontal asymptote, originally at $ y = 0 $, moves down to $ y = -2 $.

Determine Domain and Range

The domain remains all real numbers ($ x \in \mathbb{R} $) because the transformations do not restrict the $ x $-values. However, the range changes due to the reflection and the shift: all values less than $-2$. Thus, the range is $ y < -2 $.

State the Asymptote

The horizontal asymptote, originally at $ y = 0 $, has been shifted down to $ y = -2 $ due to the vertical shift in the function equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Functions

Exponential functions form a vital part of mathematics due to their unique properties and applications. An exponential function is of the form $ y = a^{x} $, where $ a $ is a constant and $ x $ is the exponent. In this formula, $ a $ is known as the base, and it must be a positive number other than 1. One of the most common exponential functions is $ y = e^x $, where $ e $ is a mathematical constant approximately equal to 2.71828. This particular function is significant in various branches of science and engineering due to its natural growth properties.

The domain of exponential functions like $ y = e^x $ is all real numbers, meaning $ x $ can be any real value.
The range of $ y = e^x $ is always positive real numbers since the function output never becomes zero or negative.
Exponential functions have unique characteristics, such as rapidly increasing or decreasing values, which can model real-world processes like population growth or radioactive decay.

Function Transformations

Function transformations involve shifting, stretching, reflecting, or compressing a function's graph without changing its basic shape. These transformations make it easier to understand relationships between different functions and their graphs. To transform the exponential function $ y = e^x $:

A horizontal shift occurs when you have $ e^{x-c} $. The graph shifts $ c $ units to the right if $ c $ is positive, and $ |c| $ units to the left if $ c $ is negative.
A vertical shift is introduced by adding or subtracting from the whole expression, like $ y = e^x + c $, moving the graph $ c $ units up or $ |c| $ units down.
Reflection over the x-axis happens with a negative sign, as in $ y = -e^x $. This flips the graph upside down.

For example, in the function $ g(x) = -e^{x-1} - 2 $, there's:

A horizontal shift right by 1 unit.
A reflection over the x-axis.
A vertical shift down by 2 units.

Asymptotes

Asymptotes refer to lines that a graph approaches but never actually touches. They are especially important in graphing exponential functions. Knowing the asymptotes of a function helps to understand its behavior at the extremes. In a basic exponential function like $ y = e^x $, there is a horizontal asymptote at $ y = 0 $. This means, regardless of how large or small $ x $ becomes, $ y $ will never actually reach zero.
In transformations, asymptotes can shift. For $ g(x) = -e^{x-1} - 2 $, like the example above:

The graph is horizontally shifted, but this does not affect the asymptote.
The vertical shift of $-2$ moves the asymptote from $ y = 0 $ to $ y = -2 $.
The reflection does not change the location of the horizontal asymptote.

Understanding asymptotes is crucial in predicting the graph's behavior, particularly when examining limits and end-behavior of functions.

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91影视

Graph the function, not by plotting points, but by starting from the graph of \(y=e^{x}\) in Figure \(1 .\) State the domain, range, and asymptote. $$g(x)=-e^{x-1}-2$$

Short Answer

Step by step solution

Identify the Base Graph

Apply Horizontal Transformation

Apply Reflection and Vertical Shift

Determine Domain and Range

State the Asymptote

Key Concepts

Exponential Functions

Function Transformations

Asymptotes

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