Given that 𝑎 is a constant, what is the domain of the function 𝑓 of 𝑥 equals the absolute value of 𝑥 plus 𝑎?
We’ll begin by recalling what we actually mean by the domain of a function. The domain of a function is the complete set of possible values of the independent variable, in other words, the set of possible 𝑥-values that make the function work and will output real 𝑦-values. So, let’s look at our function. 𝑓 of 𝑥 is equal to the absolute value of 𝑥 plus 𝑎, where 𝑎 is a constant. What that means is we substitute a value for 𝑥 into the expression 𝑥 plus 𝑎. Whatever our output, we make that positive. And so, we think, are there any values of 𝑥 that won’t make the function work? Well, no, there isn’t. In fact, the domain of our function is simply all real numbers.
And, in fact, we can say in general that a function of the form the absolute value of some polynomial will have a domain of all real numbers. And whilst the question doesn’t ask us to, we might even want to consider what the range of this function would be. Well, it’s the complete set of possible resulting values of the independent variable after we’ve substituted the domain, so the resulting 𝑦-values we get after substituting all our possible 𝑥-values.
And we said that what happens is we substitute a value for 𝑥, add some constant, and that could be a positive or negative constant. And whatever our result, we make it positive. So, we should be quite clear that our output, the resulting 𝑦-values, will always be positive.
But we can also say that if 𝑥 is actually equal to negative 𝑎, 𝑥 plus 𝑎 is zero and the absolute value of zero is zero. So, the range does include zero itself. And we can, therefore, use the set notation shown to represent the range. We know that the resulting 𝑦-values will be greater than or equal to zero and less than ∞.