Video Transcript
Simplify the function 𝑛 of 𝑥 equals 𝑥 cubed plus 343 over two 𝑥 squared plus 14𝑥 times 𝑥 plus three over 𝑥 squared minus seven 𝑥 plus 49, and determine its domain.
Inspecting 𝑛 of 𝑥 carefully and we notice, in fact, it’s the product of two rational functions, a rational function, of course, being the quotient of a pair of polynomials. So let’s remind ourselves how we find the domain when we’re working with a function that is itself the combination of two or more functions. We first remind ourselves that the domain is just the set of possible inputs to the function. And when we’re dealing with the domain of a function that’s made up of a combination of functions, that is the intersection of the domains of those respective functions. So, if we define the first fraction to be 𝑓 of 𝑥, we know we need to find the domain of these functions individually and then find their intersection to find the domain of 𝑛 of 𝑥. We’ll perform this step before we actually simplify the fraction.
So, let’s begin by thinking about the domain of 𝑓 of 𝑥. We said that 𝑓 of 𝑥 is a rational function. And we can quote that the domain of a rational function is the set of real numbers excluding any values of 𝑥 that make the denominator equal to zero. So, the domain of 𝑓 of 𝑥 is the set of real numbers excluding any values of 𝑥 that make two 𝑥 squared plus 14𝑥 equal to zero. To find such values of 𝑥, we’ll solve the equation two 𝑥 squared plus 14𝑥 equals zero. Since the left-hand side shares a common factor of two 𝑥, we factor to get two 𝑥 times 𝑥 plus seven equals zero. Then, for the product of these two expressions to be zero, we know either one or other of the expressions itself must be equal to zero. That is, two 𝑥 equals zero, meaning 𝑥 equals zero, or 𝑥 plus seven equals zero, meaning 𝑥 equals negative seven.
These are the values of 𝑥 we choose to exclude from the domain of our function. So, the domain of 𝑓 of 𝑥 is the set of real numbers minus the set containing negative seven and zero. Let’s repeat this process for 𝑔 of 𝑥. Once again, it’s a rational function. So its domain will be the set of real numbers not including those that make the denominator equal to zero. To find those values we need to exclude, we set the denominator equal to zero and solve for 𝑥. So, 𝑥 squared minus seven 𝑥 plus 49 equals zero. Now, this expression, 𝑥 squared minus seven 𝑥 plus 49, is not factorable. That’s because there are no numbers that multiply to make 49 and add to make negative seven. In fact, we can investigate this further by looking at the discriminant of this quadratic.
For a quadratic of the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero, the discriminant is 𝑏 squared minus four 𝑎𝑐. So, in this case, it’s negative seven squared minus four times one times 49. That’s negative 147. And this is a negative number. It’s less than zero. This means the equation 𝑥 squared minus seven 𝑥 plus 49 equals zero has no real solutions. There are no values of 𝑥 that make 𝑥 squared minus seven 𝑥 plus 49 equal to zero. And that’s great because there are no values of 𝑥 that we need to exclude from our domain. The domain of 𝑔 of 𝑥 is simply the set of real numbers. So, we can now identify the domain of 𝑛 of 𝑥. It’s the intersection of our two domains. In other words, it’s the set of real numbers minus the set containing negative seven and zero.
Now that we have the domain of 𝑛 of 𝑥, we’re ready to simplify. Let’s replace the denominator of 𝑓 of 𝑥 with its factored form, two 𝑥 times 𝑥 plus seven. We do this because when we simplify a function, we want to factor wherever possible. This will then allow us to identify common factors that we can divide through. We’ve already said that 𝑥 squared minus seven 𝑥 plus 49 is not factorable, nor is the expression 𝑥 plus three. But what about 𝑥 cubed plus 343? Well, in fact, we know 343 is seven cubed. So we can use the sum of two cubes formula. That is, 𝑥 cubed plus 𝑦 cubed can be written as 𝑥 plus 𝑦 times 𝑥 squared minus 𝑥𝑦 plus 𝑦 squared.
To identify how to factor 𝑥 cubed plus 343, we write it as 𝑥 cubed plus seven cubed. This means we can then use the formula to get 𝑥 plus seven over 𝑥 squared minus seven 𝑥 plus seven squared, which we can then write as 𝑥 plus seven times 𝑥 squared minus seven 𝑥 plus 49. And now we can rewrite 𝑛 of 𝑥 using the factored form of 𝑥 cubed plus 343. Now, the reason we do this is because we can now start to look for common factors that we can divide through. For instance, take 𝑥 plus seven in the numerator and denominator of our fraction. Since we excluded 𝑥 equals negative seven from our domain, we’re never dividing zero by zero, which is undefined. And so 𝑥 plus seven divided by 𝑥 plus seven will always be one.
Similarly, we cancel out the 𝑥 squared minus seven 𝑥 plus 49 factors. And we see this leaves us with one over two 𝑥 times 𝑥 plus three over one. Finally, all we need to do is multiply the fractions by multiplying their numerators and separately multiplying their denominators. And that gives us 𝑥 plus three over two 𝑥. So, 𝑛 of 𝑥 is 𝑥 plus three over two 𝑥. And our domain is the set of real numbers minus the set containing negative seven and zero.
