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Chapter 4: Problem 16

Find the domain and range of each function: $$ f(x)=\ln \left(1-x^{2}\right) $$

Short Answer

Expert verified
Domain: \((-1, 1)\); Range: \((-\infty, 0)\)

Step by step solution

01

Understand the Function

The function provided is \(f(x) = \ln(1-x^2)\). The natural logarithm function \(\ln(x)\) requires that its argument be greater than zero to be defined.
02

Determine the Domain

Since the argument of the logarithm in \(f(x)\) is \(1-x^2\), we need \(1-x^2 > 0\). This inequality can be rewritten as \(1 > x^2\). Solving this inequality gives \(-1 < x < 1\). Therefore, the domain of \(f(x)\) is \((-1, 1)\).
03

Determine the Range

The range of \(f(x)\) is dependent on the outputs of the \(\ln(y)\) function. Since \(1 - x^2\) approaches values between \(0\) and \(1\) (as \(x\) is in \((-1, 1)\)), the value of \(\ln(1-x^2)\) will vary from \(-\infty\) (as \(x\) approaches 卤1) to \(0\) (when \(x = 0\)). Thus, the range of \(f(x)\) is \((-\infty, 0)\).

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Key Concepts

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

Understanding the Natural Logarithm Function
The natural logarithm function, denoted as \( \ln(x) \), is an essential concept in mathematics, particularly in calculus and real analysis. Its base is the mathematical constant \( e \), approximately equal to 2.71828. Understanding how it works is important for solving equations and analyzing functions like \( f(x) = \ln(1-x^2) \).
  • Key Characteristics: The natural log has a domain of \( x > 0 \), meaning it only accepts positive numbers as inputs. If you attempt to take the natural log of zero or a negative number, you will get an undefined result.
  • Behavior: As \( x \) increases, \( \ln(x) \) grows steadily but at a decreasing rate, indicating a concave shape when graphed. It's valuable in many real-world contexts where exponential growth processes need to be examined.
To understand the context of this exercise, remember that the argument \( 1 - x^2 \) must be positive for the function to be defined.
This criterion sets the stage for solving inequalities to identify the suitable domain of the function.
The Art of Solving Inequalities
Solving inequalities is a fundamental skill in math that involves finding the set of values for which a given inequality holds true. When dealing with functions like \( f(x) = \ln(1-x^2) \), solving inequalities is crucial for determining domains.
  • Step-by-Step: Start by writing down the inequality derived from the requirement of the logarithm. Here, we have \( 1-x^2 > 0 \).
  • Re-arrange and Simplify: This can be rewritten as \( 1 > x^2 \), or symmetrically, \( -1 < x < 1 \). This means that the variable \( x \) can take any value between \(-1\) and \(1\) (non-inclusive).
  • Graphical Approach: Visualizing \( 1-x^2 > 0 \) can also help, as it represents the area between two points on a number line without including the endpoints where the graph of \( x^2 \) crosses the horizontal line at 1.
Understanding how to solve these inequalities ensures that we correctly identify where a function is defined, particularly for constraints imposed by operations like logarithms.
Function Analysis: Domain and Range
Analyzing functions like \( f(x) = \ln(1-x^2) \) involves understanding their behavior over their domain and identifying all possible output values (range).
  • Domain: As established, the domain of \( f(x) \) is determined by the inequality \( 1 - x^2 > 0 \), resulting in \( x \) values ranging from \(-1\) to \(1\). This ensures the function remains within defined logarithmic conditions.
  • Range: To find the range, observe the transformation of potential \( x \) values through \( 1-x^2 \) and further through the natural log. As \( x \) approaches the boundary values of -1 or 1, the expression \( 1-x^2 \) tends to zero, making \( \ln(1-x^2) \) approach \(-\infty\).
  • Outputs Across Domain: At \( x = 0 \), the value \( \ln(1 - 0^2) = \ln(1) = 0 \) is the maximum output within this function's domain.
    Therefore, \( f(x) \) produces a continuous range from \(-\infty\) to \(0\), illustrating the full behavior of outputs while maintaining defined conditions.
By piecing together these insights, students can understand not only the domains and ranges but also the mathematical reasoning that underpins analyzing such intricate functions.

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Most popular questions from this chapter

For each function: a. Find the relative rate of change. b. Evaluate the relative rate of change at the given value(s) of \(t\). $$ f(t)=e^{-t^{3}}, \quad t=5 $$

A supply function \(S(p)\) gives the total amount of a product that producers are willing to supply at a given price \(p\). The elasticity of supply is defined as $$ E_{s}(p)=\frac{p \cdot S^{\prime}(p)}{S(p)} $$ Elasticity of supply measures the relative increase in supply resulting from a small relative increase in price. It is less useful than elasticity of demand, however, since it is not related to total revenue. Use the preceding formula to find the elasticity of supply for a supply function of the form \(S(p)=a p^{n}\), where \(a\) and \(n\) are positive constants.

Derive the formula $$ a^{x}=e^{(\ln a) x} \quad(\text { for } a>0) $$ by identifying the property of natural logarithms (from the inside back cover) that justifies the first equality in the following sequence: $$ a^{x}=\left(e^{\ln a}\right)^{x}=e^{(\ln a) x} $$

For each demand function \(D(p)\) : a. Find the elasticity of demand \(E(p)\). b. Determine whether the demand is elastic, inelastic, or unit-elastic at the given price \(p\). $$ D(p)=6000 e^{-0.05 p}, \quad p=100 $$

The demand function for automobiles in a dealership is given below, where \(p\) is the selling price. a. Use the method described in the Graphing Calculator Exploration on page 312 to find the elasticity of demand at a price of $$\$ 12,000$$. b. Should the dealer raise or lower the price from this level to increase revenue? c. Find the price at which elasticity equals 1. [Hint: Use INTERSECT. $$ D(p)=\frac{200}{8+e^{0.0001 p}} $$
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