Question

Find the domain of the function. $f(x)=\frac{2 x^3-5}{x^2+x-6}$

   Find the domain of the function.
$f(x)=\frac{2 x^3-5}{x^2+x-6}$

Calculus

James Stewart 7th Edition

Chapter 1, Problem 32

Instant Answer

Solved by Expert William Semus

10/06/2023

Step 1

For a rational function like this one, we need to ensure that the denominator is not equal to zero. Show more…

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Find the domain of the function. $f(x)=\frac{2 x^3-5}{x^2+x-6}$

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Key Concepts

Excluded Values

When finding the domain of a function, especially rational functions, it is crucial to identify any values that would cause an undefined operation, particularly division by zero. This is done by setting the denominator equal to zero, solving for the variable, and then excluding those solutions from the domain.

Domain of a Function

The domain of a function is the set of all input values for which the function is defined. In any function, some operations may not be allowed for certain values (such as taking the square root of a negative number or dividing by zero), so determining the domain involves identifying these restrictions.

Rational Functions

Rational functions are functions that can be expressed as the ratio of two polynomials. They are particularly sensitive to the issue of division by zero. To determine the domain of a rational function, one must ensure that the denominator does not equal zero, as this would make the function undefined.

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