Problem 66 - Precalculus Mathematics for Calculus

Question

Find the domain of the function. $$g(x)=\ln \left(x-x^{2}\right)$$

Step-by-Step Solution

Answer

The domain is \( (0, 1) \).

Step 1: Identify the Function Type

The function given is the natural logarithm function, \( g(x) = \ln(x - x^2) \). The natural logarithm function \( \ln(u) \) is only defined for \( u > 0 \). Therefore, we need to determine the values of \( x \) that make \( x - x^2 > 0 \).

Step 2: Set the Argument Greater Than Zero

Set the argument of the logarithmic function greater than zero: \( x - x^2 > 0 \). This inequality will help us find the values for which the function \( g(x) \) is defined.

Step 3: Factor the Quadratic Expression

Factor the expression \( x - x^2 \) as \( x(1-x) \). This gives us the inequality \( x(1-x) > 0 \).

Step 4: Solve the Inequality

Analyze the inequality \( x(1-x) > 0 \). Determine the intervals where the product is positive by considering the signs of each factor:- \( x > 0 \)- \( 1 - x > 0 \)Therefore, both conditions are satisfied when \( 0 < x < 1 \).

Step 5: Conclusion on the Domain

From Step 4, we've concluded that the function is defined for \( x \) in the interval \( (0, 1) \). Thus, the domain of \( g(x) = \ln(x - x^2) \) is \( (0, 1) \).

Key Concepts

Natural Logarithm

The natural logarithm is denoted as \( \ln(x) \), where \( x \) is a positive real number. It is the logarithm to the base \( e \), an irrational constant approximately equal to 2.71828. Natural logarithms are essential in various mathematical fields and are often used in calculus, physics, and engineering.

The key property of the natural logarithm function is that it is only defined for positive real numbers. This means the domain of the natural logarithm function \( \ln(x) \) is \( x > 0 \). If you try to take the logarithm of a zero or negative number, it is undefined and does not yield real values.

Understanding this is crucial when determining the domain of a logarithm-based function like \( g(x) = \ln(x - x^2) \). The expression inside the logarithm, \( x - x^2 \), must be greater than zero for the natural logarithm to be valid.

Quadratic Inequality

A quadratic inequality involves a quadratic expression and an inequality symbol like \( > \), \( < \), \( \geq \), or \( \leq \). The inequality \( x - x^2 > 0 \) from our problem is a perfect example.

To solve such inequalities, you often need to factor the quadratic expression. For \( x - x^2 \), factoring gives \( x(1-x) \). Solving \( x(1-x) > 0 \) requires finding intervals where the product of the factors is positive.

A useful method here is the test point method:

• Identify the roots or zeros of the expression, i.e., points where \( x = 0 \) and \( x = 1 \).
• These points divide the real number line into intervals to test.
• Evaluate the product \( x(1-x) \) at a test point from each interval to determine where it is positive.

For our original exercise, the suitable interval was determined as \( 0 < x < 1 \). This means within this interval, \( x - x^2 > 0 \), satisfying the condition for the logarithm's domain.

Real Numbers

Real numbers comprise all the numbers on the continuous number line we deal with in everyday life. This includes integers, fractions, and irrational numbers like \( \pi \) and \( e \).

Real numbers are crucial in analyzing and solving equations and inequalities. When discussing the domain of a function, we often talk about real values of \( x \) that the function can accept. For example, with the function \( g(x) = \ln(x - x^2) \), the domain was determined using real numbers.

While solving mathematical problems, it's essential to note that while some functions might accept any real number, others, like logarithms, only accept positive real numbers. Inequalities such as \( x(1-x) > 0 \) are solved over real numbers, determining the intervals where solutions lie.