Question

Find the domain of the function. $$f(x)=\frac{x^{4}}{x^{2}+x-6}$$

   Find the domain of the function.
$$f(x)=\frac{x^{4}}{x^{2}+x-6}$$

College Algebra

College Algebra

James Stewart,… 7th Edition

Chapter 2, Problem 58

Instant Answer

Solved by Expert James Kiss

06/01/2020

Step 1

The given function is a rational function, which is a ratio of two polynomials. The function is \( f(x) = \frac{x^4}{x^2 + x - 6} \). Show more…

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Find the domain of the function. $$f(x)=\frac{x^{4}}{x^{2}+x-6}$$

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Key Concepts

Domain of a Function

The domain of a function is the complete set of possible input values (typically x-values) for which the function is defined. This involves identifying any restrictions that might cause the function to be undefined, such as division by zero or taking even roots of negative numbers.

Rational Functions

A rational function is an expression that represents the ratio of two polynomials. The domain of a rational function is all real numbers except those that make the denominator equal to zero, as division by zero is undefined.

Zeros of the Denominator

Identifying the zeros of the denominator is a crucial step in finding the domain of a rational function. By setting the polynomial in the denominator equal to zero and solving for the variable, one can find the values that must be excluded from the domain.

Solving Quadratic Equations

When the denominator is a quadratic polynomial, solving it to find its zeros typically involves factorization, using the quadratic formula, or completing the square. This step is essential in determining the particular values that are not part of the domain.

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