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Chapter 3: Problem 16

Find the domain of each rational function. $$f(x)=\frac{-x^{2}+2 x-3}{x^{2}+x-12}$$

Short Answer

Expert verified
The domain is all real numbers except \( x = -4 \) and \( x = 3 \).

Step by step solution

01

Identify the Rational Function

The given rational function is \( f(x) = \frac{-x^{2} + 2x - 3}{x^{2} + x - 12} \).
02

Set the Denominator to Zero

To find the domain of the rational function, we must identify values of \( x \) that make the denominator zero. Set the denominator equal to zero: \( x^{2} + x - 12 = 0 \).
03

Solve the Quadratic Equation

Solve the quadratic equation \( x^{2} + x - 12 = 0 \) for \( x \). Factor the quadratic: \( (x + 4)(x - 3) = 0 \). This gives the solutions \( x = -4 \) and \( x = 3 \).
04

Determine the Domain

The values \( x = -4 \) and \( x = 3 \) make the denominator zero, thus they are not included in the domain. Therefore, the domain of \( f(x) \) is all real numbers except \( x = -4 \) and \( x = 3 \).

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Key Concepts

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

Rational Functions
Rational functions are a type of function expressed as a fraction. The numerator and denominator are both polynomials. For example, the function given in the exercise, \( f(x) = \frac{-x^{2} + 2x - 3}{x^{2} + x - 12} \), is a rational function.

Rational functions can have values that are undefined. Specifically, these are the values where the denominator is zero. If you divide by zero, you get something undefined, and this forms key considerations in rational functions.

We need to always check the denominator of a rational function to understand its domain.
Domain of a Function
The domain of a function is all the possible input values (x-values) that you can use in the function without making it undefined.

For rational functions, the usual condition is that the denominator must not be zero. So, finding the domain means we need to find and exclude these x-values.

In the exercise, we have the rational function \( f(x) = \frac{-x^{2} + 2x - 3}{x^{2} + x - 12} \). To find the domain, we set the denominator equal to zero and solve for x: \( x^{2} + x - 12 = 0 \). The solutions to this equation will be excluded from the domain.
Quadratic Equations
Quadratic equations are polynomial equations of degree 2. They take the form \( ax^2 + bx + c = 0 \). The solutions to quadratic equations can be found using various methods such as factoring, completing the square, or the quadratic formula.

In our example, we have the quadratic equation \( x^{2} + x - 12 = 0 \) from our denominator in the rational function.

We use factoring to solve this, which leads us to the solutions \( x = -4 \) and \( x = 3 \). These solutions indicate where the denominator is zero and thus are excluded from the domain of the function.
Factoring
Factoring is a method used to solve quadratic equations by writing them as a product of linear factors. It's useful because it helps us find the roots of the equation.

For the quadratic equation \( x^{2} + x - 12 = 0 \), we factor it as \((x + 4)(x - 3) = 0 \). Setting each factor equal to zero gives us \( x = -4 \) and \( x = 3 \).

This factoring process not only simplifies solving the equation but also helps in understanding the domain of rational functions in the context of the exercise.

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