### Video Transcript

Determine the domain and range of the function ๐ of ๐ฅ is equal to the absolute value of five ๐ฅ plus five minus nine.

We begin by recalling what we actually mean by the domain and range of a function. We say that the domain of a function is the complete set of possible values for our independent variable, in other words, the set of ๐ฅ-values that make the function work and will essentially output real values for ๐ฆ. Then, the range is the complete set of all possible resulting values of the dependent variable after we substituted the domain. In other words, the range is the resulting ๐ฆ-values we get after substituting in all possible ๐ฅ-values.

Now, letโs think about what the function ๐ of ๐ฅ is telling us. We take values of ๐ฅ, substitute them into the expression five ๐ฅ plus five, and then make that positive. Then, we subtract nine. There are no values of ๐ฅ which donโt actually make the function work, so the domain of our function is simply all real numbers.

But what about the range? Well, thereโs two ways we can calculate this. Firstly, letโs just think about it algebraically. If we have the absolute value of five ๐ฅ plus five, what do we get when we substitute values of ๐ฅ in? Well, if ๐ฅ is equal to negative one, we get five times negative one plus five, which is zero. And the absolute value of zero is simply zero. For all other real values of ๐ฅ, we get a result thatโs greater than zero. And so, we can say that the range of the function the absolute value of five ๐ฅ plus five is greater than or equal to zero.

Weโre going to subtract nine from both sides of this inequality. And we find that the absolute value of five ๐ฅ plus five minus nine must be greater than or equal to negative nine. And so, using set notation for the range, we find itโs greater than or equal to negative nine and less than โ.

But there is another way we can find the range. And thatโs to consider the graph of the function. We take the graph of ๐ฆ equals five ๐ฅ plus five. Itโs a single straight line that passes through the ๐ฆ-axis at five and the ๐ฅ-axis at negative five. We then find the absolute value of all of our outputs, of all of our values of ๐ฆ. In other words, any values on the graph that lie below the ๐ฅ-axis get reflected in the ๐ฅ-axis, as shown. Notice that this does include ๐ฆ equals zero.

By subtracting nine from our function, we translate it down by nine units. The lowest point on our graph has a coordinate of negative five, negative nine. And we said that the range is all possible values of ๐ฆ after weโve substituted our possible values of ๐ฅ in. We see from our graph that ๐ฆ is always greater than or equal to negative nine. And so, we end up with the same range as before. Itโs greater than or equal to negative nine and less than โ.