Video Transcript
Find the domain and range of the
function 𝑓 of 𝑥 equals seven to the power of 𝑥 plus five plus five in the set of
real numbers.
Let’s begin by reminding ourselves
what we mean by the domain and range of a function. The domain is essentially the set
of possible values that we can input to that function, in other words, the set of
𝑥-values that we can substitute into the function 𝑓 of 𝑥. Then the range is the set of
possible outputs from the function after the values from the domain have been
substituted in. Now we have in particular an
exponential function here. And the most general form of the
exponential function is 𝑔 of 𝑥 equals 𝑏 to the power of 𝑥. 𝑏 is a real number greater than
zero and not equal to one.
So we’re going to begin by
identifying what the graph of the function 𝑔 of 𝑥 looks like. This will allow us to identify its
domain and range. Then we can look at the series of
transformations that map 𝑔 of 𝑥 onto 𝑓 of 𝑥, and that will give us the graph of
the function 𝑓 of 𝑥. Then we can find its domain and
range. In particular, we’re going to begin
by sketching the graph of 𝑔 of 𝑥 equals seven to the power of 𝑥.
Since the base seven is a number
greater than one, then we know we have exponential growth instead of exponential
decay. The graph looks a lot like
this. It passes through the 𝑦-axis at
one, and it has a horizontal asymptote given by the line 𝑦 equals zero, or the
𝑥-axis. It also passes through the point
one, seven. Now we can learn this fact, or we
can substitute 𝑥 equals one into the function 𝑔 of 𝑥 and we do indeed get
seven.
With that in mind, let’s identify
what the domain and range of the function 𝑔 of 𝑥 is. The domain is the set of possible
values we can input into the function. We can think about this as the
spread of values in the 𝑥-direction. There are no limits here. We can substitute any 𝑥-value into
the function 𝑔 of 𝑥 and we will always get a real output. The range is a little bit different
though. The range can be thought of as the
spread of values in the 𝑦-direction. And we said that there was a
horizontal asymptote given by the line 𝑦 equals zero. This means the function 𝑔 of 𝑥
approaches zero but never quite reaches it. In the other direction though, it
heads off towards positive ∞. And so the range is the open
interval from zero to positive ∞.
With this in mind, how do we map
the function 𝑔 of 𝑥 equals seven to the power of 𝑥 onto the function 𝑓 of
𝑥? Let’s begin by mapping seven to the
power of 𝑥 onto seven to the power of 𝑥 plus five. This is a horizontal translation of
the function 𝑔 of 𝑥 five units to the left or by the vector negative five,
zero. Then what happens if we add five to
the function seven to the power of 𝑥 plus five? Well, this is a vertical
translation five units up or by the vector zero, five. So to map the function 𝑔 of 𝑥
equals seven to the power of 𝑥 onto the function 𝑓 of 𝑥, we’re going to perform
two translations, or we can think about it as one big translation. It’s going to move five units left
and then five units up.
This time we know it will have a
horizontal asymptote five units above the previous one. Since the previous one was given by
the line 𝑦 equals zero, the new asymptote will have the equation 𝑦 equals
five. And whilst not entirely necessary,
we can find the value of the 𝑦-intercept by substituting 𝑥 equals zero into the
function. When we do, we get seven to the
fifth power plus five. And we can also see that this
passes through the point one, seven to the power of six plus five. That’s found by substituting one
into the function.
We now, however, have enough
information to determine the domain and range of this new function. Once again, the domain can be
thought of as the spread of values in the 𝑥-direction. This is once again unbounded. So the domain of 𝑓 of 𝑥 is the
set of real numbers. The range once again can be thought
of as the spread of values in the 𝑦-direction. This time the horizontal asymptote
is given by the line 𝑦 equals five, so 𝑓 of 𝑥 approaches five but never quite
reaches it. In the other direction, it heads
off towards positive ∞. So the range of the function is the
open interval from five to ∞.
And so by considering graphs and
transformations of graphs, we’ve determined the domain and range of the function 𝑓
of 𝑥 equals seven to the 𝑥 plus fifth power plus five. The domain is the set of real
numbers, and the range is the open interval from five to ∞.
