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Chapter 3: Problem 55

Find the domain of each function. \(g(x)=\frac{x}{x^{2}-16}\)

Short Answer

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

Step by step solution

01

Identify the function and its denominator

The given function is \( g(x) = \frac{x}{x^2 - 16} \). To find the domain, look at the denominator \( x^2 - 16 \). The denominator must not be equal to zero, because division by zero is undefined.
02

Set the denominator to zero

Set the denominator \( x^2 - 16 \) to zero and solve for \( x \). This means solving the equation: \[ x^2 - 16 = 0 \].
03

Factor the quadratic equation

Factor the quadratic equation \( x^2 - 16 = 0 \) using the difference of squares: \[ (x - 4)(x + 4) = 0 \].
04

Solve for \( x \)

Set each factor equal to zero and solve for \( x \):\[ x - 4 = 0 \ x = 4 \] and \[ x + 4 = 0 \ x = -4 \].
05

Determine the domain

The values \( x = 4 \) and \( x = -4 \) cause the denominator to be zero, so they must be excluded from the domain. Therefore, the domain of the function \( g(x) = \frac{x}{x^2 - 16} \) is all real numbers except \( x = 4 \) and \( x = -4 \).

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Key Concepts

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

Rational Functions
A rational function is a type of function that is the ratio of two polynomials. The general form of a rational function is \( f(x) = \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) \) is not zero. Rational functions can have points where they are not defined because the denominator cannot be zero. Identifying these points is crucial to find the domain. In our exercise, the function \( g(x) = \frac{x}{x^2 - 16} \) is a rational function. To find its domain, we look at its denominator \(x^2 - 16\) and determine when it equals zero.
Quadratic Equations
Quadratic equations are equations of the form \( ax^2 + bx + c = 0 \) where \( a, b, \) and \( c \) are constants. In our problem, we need to solve the quadratic equation \( x^2 - 16 = 0 \) to find values of \( x \) that make the denominator zero. Quadratic equations can be solved using various methods, such as factoring, completing the square, or using the quadratic formula. Here, we use factoring due to the specific nature of our quadratic equation.
Difference of Squares
The difference of squares is a mathematical identity that says \( a^2 - b^2 = (a - b)(a + b) \). This identity is useful for factoring quadratic equations like the one in our exercise. For \( x^2 - 16 = 0 \), we recognize it as a difference of squares where \( x^2 \) is \( a^2 \) and \( 16 \) is \( 4^2 \). Thus, we can factor it as \( (x - 4)(x + 4) = 0 \). By solving each factor set to zero, we find that the solutions are \( x = 4 \) and \( x = -4 \). These values are excluded from the domain of our function \( g(x) \).

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