Video Transcript
Using the graph, find the
values of 𝑥 for which 𝑓 of 𝑥 is greater than or equal to zero. And we’re given that 𝑓 of 𝑥
is equal to minus 𝑥 squared plus five 𝑥. And we’re also given the graph
which shows 𝑦 equals 𝑓 of 𝑥, which actually maps out that quadratic. So this is a symmetrical
parabola, cuts the 𝑥-axis at zero and five, and also cuts the 𝑦-axis at
zero. So we’ve gotta find out when 𝑓
of 𝑥 is greater than zero; in other words, when the 𝑦-coordinate is greater
than or equal to zero.
So with 𝑦 equals 𝑓 of 𝑥, and
we’re looking for when 𝑓 of 𝑥 is greater than or equal to zero, we’re really-
we’re looking for are all the points on that curve where the 𝑦-coordinate is
greater than or equal to zero. Well here at zero and here at
five on the 𝑥-axis, the 𝑦-coordinate is zero. And on all these points on the
curve in between, the 𝑦-coordinate is greater than zero. So that’s the region that we’re
interested in. So we wanna know the
𝑥-coordinates that generate those. Well zero, five, everything in
between in terms of the 𝑥-coordinate, so from zero to five, and everything
outside that region. So bigger than five onwards or
less than zero onwards down to negative infinity, that’s not included in our
region because these parts of the curve are not greater than or equal to
zero. So we could write that like
this, zero is less than or equal to 𝑥 is less than or equal to five.
But we could also write it in
interval format. So the critical values are zero
and five. That’s either end of the
interval. Now zero is included, so we
need to clu- include a square bracket at the end. And five is included, so we put
the square brackets around that end. So that’s in interval
format. So we could also put it in set
notation. So we’ve got the set of 𝑥 such
that 𝑥 is a real number between zero and five. So the process then was to have
a look at the graph and identify all the points on the graph which match the
criteria that we’re looking for. In this case, it was 𝑓 of 𝑥
is greater than or equal to zero. So it’s then a question of
finding which 𝑥-coordinates match the criteria and which 𝑥-coordinates don’t
match that criteria, and then just kind of summarising that in one of these
formats: the appropriate format whether it’s the inequality format, the interval
format, or the set format depending on what the question asks for.
