The function 𝑓 of 𝑥 equals eight 𝑥 minus 𝑏 and the function 𝑔 of 𝑥 equals two 𝑥 squared minus 𝑏. Find 𝑓 of negative five plus 𝑔 of negative 10 given 𝑓 of negative 10 plus 𝑔 of negative six is equal to negative 14.
So, first of all, to help us work this out, what we need to do is find out what 𝑏 is. And we can do that using the information that we’re given. And that is that 𝑓 of negative 10 plus 𝑔 of negative six is equal to negative 14. So, the first thing we do is we substitute in negative 10 for 𝑥 in 𝑓 of 𝑥. And that’s because 𝑓 of negative 10, this is what this means.
So when we do that, we get eight multiplied by negative 10 minus 𝑏. And then we do the same with 𝑔 of 𝑥 because what we do is we substitute in negative six for 𝑥. So, we get two multiplied by negative six squared minus 𝑏. And then this is equal to minus 14.
So, now what we need to do is simplify this. And when we do that, we get negative 80, that’s cause eight multiplied by negative 10 is negative 80, minus 𝑏 plus 72 minus 𝑏 equals negative 14. And we got positive 72 because negative six squared is negative multiplied by a negative, which gives us positive. And which is 36, and two lots of 36 is 72. So, now if we tidy this up, we get negative eight minus two 𝑏 equals negative 14.
So, now if we add eight to each side of the equation, we get negative two 𝑏. And that’s cause negative eight add eight is zero. So, we get negative two 𝑏 is equal to negative six. So then, we divide by negative two and leaves us with 𝑏 is equal to three. Okay great, so we found 𝑏. So, now what we can do is look to solve the problem. And the problem asks us to find 𝑓 of negative five plus 𝑔 of negative 10. So, what we can do is substitute in the values we now know.
So, when we substitute our values for 𝑥, so negative five for 𝑥 in 𝑓 of 𝑥 and negative 10 for 𝑥 in 𝑔 of 𝑥, we get eight multiplied by negative five minus three plus two multiplied by negative 10 squared minus three. And this is gonna give us negative 40 minus three plus 200 minus three, which is 154. And that’s because if we have 200 and we subtract 40, we get 160 subtract three gives us 157. And then subtract three more gives us 154. So finally, we can say that 𝑓 of negative five plus 𝑔 of negative 10 is equal to 154.