Problem 80 - Precalculus with Calculus Previews

Question

(a) Use a graphing utility to obtain the graph of the function \(f(x)=\ln \left(x+\sqrt{x^{2}+1}\right)\) (b) Show that \(f\) is an odd function, that is, \(f(-x)=\) \(-f(x)\)

Step-by-Step Solution

Answer

The function is odd: its graph is symmetric about the origin; algebraically, \(f(-x) = -f(x)\).

Step 1: Graph the Function

Use a graphing utility such as Desmos or a graphing calculator to plot the function \(f(x) = \ln(x+\sqrt{x^2+1})\). Observe that the graph is symmetric about the origin, which is characteristic of odd functions.

Step 2: Verify the Odd Function Property Algebraically

To verify that \(f(x)\) is odd, calculate \(f(-x)\) and show that it equals \(-f(x)\). Start by substituting \(-x\) into the function:\[ f(-x) = \ln(-x+\sqrt{(-x)^2+1}) = \ln(-x + \sqrt{x^2+1}). \]

Step 3: Simplify and Compare

Notice that the inside of the logarithm can be rewritten using properties of logarithms:\[ f(-x) = \ln((-1)(x + \sqrt{x^2+1})) = \ln(-1) + \ln(x+\sqrt{x^2+1}). \]Since \(\ln(-1)\) is undefined for real numbers, instead, use symmetry property:Observe \(f(x) + f(-x) = \ln(x+\sqrt{x^2+1}) + \ln(-x+\sqrt{x^2+1}) = \ln((x+\sqrt{x^2+1})(-x+\sqrt{x^2+1}))\).

Step 4: Continue Simplifying

Continue simplifying the expression obtained from Step 3:\[\ln((x+\sqrt{x^2+1})(-x+\sqrt{x^2+1})) = \ln((x-x+1) = \ln(1).\]Since \(\ln(1) = 0\), it follows that\[ f(x) + f(-x) = 0, \text{ or equivalently, } f(-x) = -f(x). \] This confirms the function is odd.

Key Concepts

Graphing Utility

A graphing utility is a tool that helps visualize mathematical functions and their properties. These tools, like Desmos or graphing calculators, allow you to input functions and see their graphical representation.
For the given function \(f(x) = \ln(x + \sqrt{x^2+1})\), using a graphing utility can show you how the function looks.
When you graph this function, you'll notice it is symmetric about the origin, a key characteristic of odd functions.

• Graphing utilities make complex calculations visually understandable.
• They help confirm algebraic results with visual evidence.
• They provide insights into function behaviors and properties such as symmetry and periodicity.

By seeing the graph, it becomes evident that this function mirrors itself horizontally and vertically across the origin, which is instrumental in identifying it as an odd function.

Logarithmic Function

Logarithmic functions are functions of the form \(f(x) = \ln(x)\), representing the natural logarithm, which is the inverse of the exponential function. These functions have particular properties:

• Defined only for positive real numbers.
• They grow logarithmically, meaning they increase but at a decreasing rate as \(x\) increases.
• Their domain is \(x > 0\), and their range is all real numbers.

The function \(f(x) = \ln(x + \sqrt{x^2+1})\) modifies the basic logarithmic expression by incorporating a square root addition inside.
This affects its domain and range while maintaining core properties of logarithmic functions.
Understanding the behavior of basic logarithmic functions helps analyze more complex functions quickly, identifying transformations and their effects on the graph.

Properties of Functions

Functions have different properties that help describe their behavior, such as symmetry, continuity, and periodicity.

• Odd Functions: Functions symmetrical about the origin where \(f(-x) = -f(x)\). This symmetry means that if you rotate the graph 180 degrees around the origin, it will look the same.
• Even Functions: Symmetrical about the y-axis where \(f(-x) = f(x)\).
• Continuity: Functions that are smooth without breaks or holes.

Identifying whether a function is odd specifically involves verifying that \(f(-x) = -f(x)\).
For \(f(x) = \ln(x + \sqrt{x^2+1})\), algebraically showing \(f(-x) = -f(x)\) confirms its odd nature.
This verification complements the visual evidence obtained through graphing, offering a complete understanding of the function’s symmetrical property.