Determine the domain and range of the function 𝑓 of 𝑥 equals 𝑥 plus seven to the sixth power over the absolute value of 𝑥 plus seven to the seventh power.
We’ll begin by recalling a few key definitions. Firstly, the domain of a function is the set of values of 𝑥, in other words, the input, which yields real outputs of the function. Then when we substitute the values of 𝑥 in from our domain, the range is the set of outputs.
Next, we recall that if we’re working with the quotient of two functions, the domain of the resulting function is the intersection of their respective domains. But we need to exclude any values of 𝑥 that make the denominator equal to zero. We have two polynomials. And we have the absolute value of the polynomial on the denominator. We know that the domain of a polynomial is a set of real numbers. And so our domain is going to be the set of real numbers excluding any values of 𝑥 that make the denominator equal to zero.
And so let’s set that denominator equal to zero and solve for 𝑥. We’re not really worried about the absolute value signs at this moment since the absolute value of zero is zero. So we ask ourselves, what values of 𝑥 make the expression 𝑥 plus seven to the seventh power equal to zero? Well, let’s find the seventh root of both sides. And we get 𝑥 plus seven equals zero. And then we subtract seven from both sides. So 𝑥 equals negative seven makes the denominator of our fraction equal to zero. And this, therefore, means that the domain of the function is the set of real numbers not including negative seven. But what do we do about the range?
Well, we know that the input is any real number not including negative seven. And so let’s think about the output of our denominator and our numerator separately. If 𝑥 is not equal to negative seven, then our denominator, the absolute value of 𝑥 plus seven to the seventh power, will never be equal to zero. The absolute value signs in fact tell us that it will always be greater than zero. And so we will always output a positive nonzero result for the denominator of our fraction. But what about 𝑥 plus seven to the sixth power?
The 𝑥 plus seven bit can be any value apart from zero according to our domain. However, when we raise any real number to an even power, we know that we output a positive result. And so the numerator of our fraction will also be a positive nonzero number. And so we’re always going to have a positive nonzero divided by another positive nonzero. And the output will always be positive nonzero. There’s no upper limit on this. So the range of our function 𝑓 of 𝑥 is the open interval from zero to ∞.