Problem 103 If a function is defined by an e... [FREE SOLUTION]

Chapter 1: Problem 103

If a function is defined by an equation, explain how to find its domain.

Short Answer

Expert verified

To find the domain of a function defined by an equation, identify the type of function and then determine the constraints or values of x for which the function is undefined. Exclude these values when writing the domain.

Step by step solution

Understand the function

Identify the type of function you're dealing with. Things to look out for may include: algebraic functions, square root functions, logarithmic functions, and rational functions. They each have a unique set of values for which they are undefined.

Identify the constraints

For algebraic functions, they're defined for all real values. For square root functions, the input must always be positive, including zero. In other words, the expression under the square root sign must be greater than or equal to zero. For rational functions which are given by a fraction, the denominator cannot be zero. So, solve the equation when the denominator equals zero to find the x-values to exclude from the domain. For logarithmic functions, the input must be greater than zero. Hence, solve the inequality when the function is greater than zero.

Write the domain

After identifying the constraints, write the domain either in interval notation, inequality notation, or in words. For example, if no real numbers need to be excluded from the domain, you could write: 'The domain is all real numbers.' If you needed to exclude certain values, you could write: 'The domain is all real numbers except...' followed by the values to be excluded.

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Key Concepts

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

Algebraic Functions

Algebraic functions are the core building blocks of many mathematical expressions. They include polynomials like linear, quadratic, and cubic functions. The domain of an algebraic function is typically all real numbers since there's often no limitation on the input. For example, in the function \( f(x) = 2x + 3 \), you can plug any real number into \( x \) and get a valid output. Here's a quick guide:

All polynomials: Domain is all real numbers \((-\infty, \infty)\).
No special constraints or exclusions.

Think of algebraic functions as the "easy-going" group of functions where the domain is almost always unrestricted.

Rational Functions

Rational functions are like fractions, where the numerator and denominator are both polynomials. The domain of a rational function is affected by the denominator, which cannot be zero. If it is, the function becomes undefined. Thus, it's crucial to find values that make the denominator zero and exclude them:

Identify the denominator of the function.
Solve for when the denominator equals zero.
Exclude these \(x\)-values from the domain.

For example, in \( f(x) = \frac{1}{x-2} \), the denominator \( x-2 = 0 \) at \( x = 2 \). Hence, the domain is all real numbers except \( x = 2 \). Use notation: \( (-\infty, 2) \cup (2, \infty) \).

Square Root Functions

Square root functions involve the square root of an expression, and their domain is restricted to ensure the value inside the root remains non-negative. Negative numbers under a square root aren鈥檛 defined in the real number system. To find the domain:

Set the expression inside the square root \( \geq 0 \).
Solve the inequality for \( x \).

For example, with \( f(x) = \sqrt{x+3} \), set \( x+3 \geq 0 \) which simplifies to \( x \geq -3 \). The domain is all real numbers \( x \geq -3 \), or in interval notation, \( [-3, \infty) \).

Logarithmic Functions

Logarithmic functions have their own unique domain rules. The input to a logarithmic function must be positive, as logarithms of zero or negative numbers are undefined. To ensure this:

Set the expression inside the logarithm \( > 0 \).
Solve the inequality to find valid \( x \)-values.

For example, with \( f(x) = \log(x-1) \), you solve \( x-1 > 0 \), leading to \( x > 1 \). Therefore, the domain is \( x > 1 \), or \( (1, \infty) \) in interval notation.

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91影视

If a function is defined by an equation, explain how to find its domain.

Short Answer

Step by step solution

Understand the function

Identify the constraints

Write the domain

Key Concepts

Algebraic Functions

Rational Functions

Square Root Functions

Logarithmic Functions

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