Video Transcript
Find the value of 𝑘 given 𝑓 of 𝑥
equals 𝑘𝑥 plus 13 and 𝑓 of eight is equal to negative 11.
Here is our function. 𝑓 is sort of the name of the
function. And 𝑥 is what goes in; it’s its
input. We’ve also been given some
information about the value of 𝑓 of eight. In other words, our input is no
longer 𝑥. It’s eight. And when the input is eight, the
output is negative 11. So, let’s see what happens when we
put eight into our definition of the function here.
𝑓 of eight means each time we see
the 𝑥, we replace or substitute it with eight. So, 𝑓 of eight becomes 𝑘 times
eight plus 13. Let’s write that as eight 𝑘 plus
13. But of course, we know that 𝑓 of
eight is equal to negative 11. So, this must mean that eight 𝑘
plus 13 must be equal to negative 11. And so, our job, we’re trying find
the value of 𝑘, is to solve this equation.
We’ll do this by performing a
series of inverse operations. The first thing we’re going to do
is subtract 13 from both sides of our equation. Remember, eight 𝑘 plus 13 minus 13
is just eight 𝑘. And negative 11 minus 13 means we
move 13 spaces further down the number line. And we get to negative 24. So, eight 𝑘 is equal to negative
24.
The eight is multiplying the
𝑘. And so, the inverse operation we
apply next is to divide both sides of our equation by eight. That leaves 𝑘 on the left-hand
side. And since 24 divided by eight is
three, we know negative 24 divided by eight is negative three. And so, given 𝑓 of 𝑥 is equal to
𝑘𝑥 plus 13 and 𝑓 of eight is equal to negative 11, we find 𝑘 is equal to
negative three.
Now, we can check our answer by
letting 𝑓 of 𝑥 now be equal to negative three 𝑥 plus 13. We’ve replaced 𝑘 with negative
three. We’re going to check that the value
of 𝑓 of eight is indeed negative 11. And so, we replace 𝑥 with
eight. And we get negative three times
eight plus 13. That’s negative 24 plus 13, which
is negative 11 as required.
