Problem 5 - Algebra and Trigonometry

Question

Find the domain of each rational function. $$ h(x)=\frac{x+7}{x^{2}-49} $$

Step-by-Step Solution

Answer

The domain of the function \(h(x)=\frac{x+7}{x^{2}-49}\) is all real values of \(x\) except \(x=7\) and \(x=-7\). In interval notation, this is \((-\infty, -7) \cup (-7, 7) \cup (7, \infty)\)

Step 1: Setting the denominator equal to zero

Identify the denominator of the rational function which is \(x^{2}-49\). Set this denominator equal to zero to find the values that will make the function undefined. \n So, solve the following equation for \(x\):\n\(x^{2}-49=0\)

Step 2: Solving the equation

To solve the equation \(x^{2}-49=0\), we can add 49 to both sides of the equation which gives us:\n\(x^{2} = 49\).\n Taking the square root of both sides, we get two possible solutions: +7 and -7.\nSo this gives us \(x = 7\) and \(x = -7\).

Step 3: Identify the domain of the function

Now, remembering that these are the values for which the function is undefined, we can state that all other values of \(x\) are the domain of the function \(h(x)\). Therefore the domain of the function \(h(x)=\frac{x+7}{x^{2}-49}\) is all real numbers except \(x = 7\) and \(x = -7\). This can be represented in interval notation as: \n\((-\infty, -7) \cup (-7, 7) \cup (7, \infty)\)