Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 5: Problem 15
Graph the function and specify the domain, range, intercept(s), and asymptote. $$y=e^{x}+1$$
Short Answer
Expert verified
Domain: (-鈭�, 鈭�); Range: (1, 鈭�); y-intercept: (0, 2); Asymptote: y=1.
Step by step solution
01
Understand the Function
The given function is \(y = e^x + 1\). This function involves the natural exponential function \(e^x\), which is shifted upward by 1 unit. The base of the natural exponential function \(e\) is approximately 2.718.
02
Determine the Domain
The domain of any exponential function is all real numbers because \(e^x\) is defined for every real number \(x\). Thus, the domain for the function \(y = e^x + 1\) is \((-\infty, \infty)\).
03
Determine the Range
The range of \(e^x\) is \((0, \infty)\) since the function never reaches negative values. With the transformation \(e^x + 1\), the entire graph is shifted up by 1 unit, changing the range to \((1, \infty)\).
04
Identify the Intercept(s)
Intercepts are points where the graph crosses the axes. For the y-intercept, set \(x = 0\): \(y = e^0 + 1 = 1 + 1 = 2\), so the y-intercept is \((0, 2)\). The function has no x-intercept as it never crosses the x-axis (since \(y\) never reaches 0).
05
Determine the Asymptote
An asymptote is a line that the graph approaches but never touches. For \(y = e^x + 1\), as \(x\) approaches negative infinity, \(y\) approaches 1, indicating a horizontal asymptote at \(y = 1\).
06
Graph the Function
Plot the graph starting by drawing the horizontal asymptote \(y = 1\). Plot the y-intercept at \((0, 2)\). For positive \(x\)-values, the graph increases exponentially, and for negative \(x\)-values, it approaches the asymptote from above.
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.
Domain of a Function
The domain of a function tells us the possible values of the independent variable, typically denoted as \(x\). For the function \(y = e^x + 1\), we need to determine for which values of \(x\) the function is defined.
Exponential functions, like \(e^x\), are defined for all real numbers. This is because you can raise the constant \(e\) (approximately 2.718) to the power of any real number and still get a valid result. Therefore, nothing restricts the input \(x\) for this function.
Exponential functions, like \(e^x\), are defined for all real numbers. This is because you can raise the constant \(e\) (approximately 2.718) to the power of any real number and still get a valid result. Therefore, nothing restricts the input \(x\) for this function.
- In other words, \(x\) can be any real number.
- \((-fty, fty)\).
Range of a Function
The range of a function is all the possible values that the function can output. For exponential functions such as \(e^x\), the values start getting interesting. Let's explore this!
- The basic form \(e^x\) has a range of \((0, fty)\). This means it never outputs negative values and never actually equals zero.
- For \(y = e^x + 1\), each output of \(y = e^x\) is simply increased by 1. This shifts the entire range upward by 1 unit.
- \((1, fty)\).
Intercepts
Intercepts are points where the graph of a function crosses the axes.
- Y-Intercept: To find where the graph crosses the y-axis, we set \(x = 0\). For \(y = e^x + 1\), we get \(y = e^0 + 1\) which simplifies to \(y = 2\). So, the y-intercept is at the point \((0, 2)\).
- X-Intercept: This point occurs where the graph crosses the x-axis. It's found by setting \(y = 0\). However, since the expression \(e^x + 1\) is always greater than 1, it never reaches 0. Hence, there are no x-intercepts.
Asymptotes
Asymptotes are invisible lines that the function approaches but never touches. They guide us to understand the behavior of a function at the extremes.
For \(y = e^x + 1\):
For \(y = e^x + 1\):
- Vertical Asymptotes: Exponential functions don't have vertical asymptotes. They occur in fractions where there's a number divided by \(x\), which doesn't apply here.
- Horizontal Asymptotes: The line \(y = 1\) is a horizontal asymptote. As \(x\) approaches negative infinity, the value of \(y = e^x\) approaches 0, making \(y = 1\) the closest line the graph approaches but never quite reaches.
