Video Transcript
Consider the function 𝑓 of 𝑥
equals the square root of 𝑥 minus one. Which of the following graphs could
represent 𝑓 of 𝑥? Using the graph of 𝑓 of 𝑥, find
its domain. Using the graph of 𝑓 of 𝑥, find
its range.
In order to identify the correct
graph of 𝑓 of 𝑥 equals the square root of 𝑥 minus one, let’s remind ourselves
what the function the square root of 𝑥 looks like when graphed. It’s the inverse of the function 𝑓
of 𝑥 equals 𝑥 squared, but we restrict it to make sure that it’s one to one. And so it looks a little something
like this. In order to use this graph to
identify the correct graph of the square root of 𝑥 minus one, let’s recall a
function transformation.
Suppose we have the graph of 𝑦
equals 𝑓 of 𝑥. This is mapped onto the graph of 𝑦
equals 𝑓 of 𝑥 minus 𝑎 for some real constant 𝑎 by a transformation by the vector
𝑎, zero. In other words, the graph is moved
𝑎 units to the right. So in this case, we are subtracting
one from the 𝑥 inside the square root symbol, meaning that the graph of 𝑔 of 𝑥
must be translated one unit to the right to map onto 𝑓 of 𝑥. So it will intersect the 𝑥-axis at
one and look a little something like this. If we look carefully at all five of
our graphs, we can observe that that is graph (B).
The next part of this question asks
us to use the graph to find the domain of our function. And so we recall that the domain is
a set of possible inputs to the function. In other words, for some function
𝑓 of 𝑥, what 𝑥-values can we substitute into that function to get real
outputs? With this definition in mind, it
follows that we can use the spread of 𝑥-values on our graph to find the domain of
our function. Now, if we look at the spread of
𝑥-values on graph (B), we see that they start at 𝑥 equals one and extend to
positive ∞. So 𝑥 can take values greater than
or equal to one. In set notation, we say that the
domain is the set of values in the left-closed, right-open interval from one to
∞.
And so we’re ready for the final
part of this question. It asks us to find the range by
using the graph of 𝑓 of 𝑥. The range of the function is the
set of possible outputs to the function when the values from the domain are
substituted in. In other words, given a function 𝑦
equals 𝑓 of 𝑥, the range is the set of possible 𝑦-values. And so we can look at the spread of
possible 𝑦-values in the vertical direction on our graph to establish the
range. On graph (B), we see that the
𝑦-values start at zero and they extend towards ∞. Now, whilst it might look like they
reach some sort of limit, we know that this is not true since the square root of ∞
minus one is simply ∞.
So the range, the set of possible
outputs, is all values greater than or equal to zero. Using set notation, the range of
the function is the left-closed, right-open interval from zero to ∞.
