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Chapter 1: Problem 36

Sketch the graph of the function and state its domain. $$ f(x)=2+\ln x $$

Short Answer

Expert verified
The graph of \(f(x)=2+\ln x\) is the graph of \(y=\ln x\), shifted upward by 2 units. The domain is \(x>0\).

Step by step solution

01

Understand the Function

First, understand that \(f(x)=2+\ln x\) is a logarithmic function. The base of the logarithm is the mathematical constant e, approximately equal to 2.71828. Logarithmic functions are the inverses of exponential functions, and the general form for a natural logarithmic function is \(f(x) = \ln x\). When a constant is added (in this case 2), it results in a vertical shift of the graph. The positive constant shifts the graph upward.
02

Identify the Domain

The natural logarithm, \(\ln x\), is only defined for positive real numbers. Therefore, \(f(x)=2+\ln x\) will only be defined for \(x > 0\). This is the domain of the function.
03

Sketch the Graph

Start by sketching \(y=\ln x\), which is a curve that gets arbitrarily close to the y-axis but never touches or crosses it (the y-axis is a vertical asymptote), and which passes through the point (1,0). Each point on \(y=\ln x\) is shifted upward by 2 units to create the graph of \(f(x)=2+\ln x\). Also, mark the vertical asymptote x=0, which is a vertical line through x=0, where the function is undefined.

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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 refers to all the possible input values (x-values) that the function can accept. For the function \(f(x) = 2 + \ln x\), understanding the domain is crucial to knowing where the function exists and is valid.

Because \(\ln x\), or the natural logarithm of \(x\), only exists for positive values of \(x\), the domain of this function is limited to \(x > 0\). This means:
  • The function cannot accept zero or negative numbers as inputs because the logarithm of these numbers is undefined.
  • In interval notation, the domain is expressed as \((0, \infty)\).
It's essential to remember that unlike basic algebraic functions, which often have no restrictions, logarithmic functions have specific domains that need careful consideration.
Natural Logarithm
The natural logarithm, denoted by \(\ln x\), is a special logarithmic function with the base \(e\), where \(e\) is approximately 2.71828. It is routinely used in mathematics due to its unique properties that relate closely with calculus and exponential functions.

Key points about natural logarithms include:
  • For \(x > 0\), \(\ln x\) provides the power to which \(e\) must be raised to obtain \(x\).
  • The function \(\ln x\) grows slowly and is continuous for positive \(x\).
  • It simplifies many problems involving exponential growth and decay, which occur naturally in the real world.
When graphed, \(\ln x\) starts from negative infinity as \(x\) approaches zero and increases without bound as \(x\) increases. It's a vital concept for understanding various applications in science and engineering.
Graph Sketching
Graph sketching involves visualizing the function on a coordinate plane, helping you understand its behavior and characteristics. For \(f(x) = 2 + \ln x\), graph sketching helps solidify how the function looks and behaves.

Here鈥檚 how to sketch its graph:
  • Start with the base curve \(y = \ln x\).
  • Note that \(y = \ln x\) has a vertical asymptote at \(x = 0\) since it never touches or crosses this line.
  • The curve passes through the point (1,0) and increases slowly to the right.
  • For \(f(x) = 2 + \ln x\), shift every point on the graph \(y = \ln x\) vertically up by 2 units.
Consequently, \(f(x) = 2 + \ln x\) still has a vertical asymptote at \(x = 0\) and shows the same slow increasing trend as \(y = \ln x\), but the entire graph is lifted upwards. This understanding helps in predicting function values and estimating the behavior in mathematical modeling.

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Most popular questions from this chapter

What is meant by an indeterminate form?

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