Find the value of 𝑓 of 14 given the function 𝑓 of 𝑥 equals 𝑥 squared minus 28𝑥 plus seven.
So we have a function here given by an algebraic rule 𝑓 of 𝑥 equals 𝑥 squared minus 28𝑥 plus seven, and that tells us for any input value 𝑥 what output value we should expect. Okay great, so what do we want to find out? We want to find out what 𝑓 of 14 is; so if we give an input of 14, what output do we get?
So for the first line of working, I’ve just written down the function again. But wherever I’ve seen an 𝑥, I’ve written that in green instead of the normal blue. Okay so here’s what we’re expected to find: 𝑓 of 14. And we’ll notice that I’ve written the 14 in orange just to make it easier to contrast with the line above to compare to it. So what’s changed from the left-hand side of the line above, 𝑓 of 𝑥, to the line line below, 𝑓 of 14? Well the 𝑥 has just changed to a 14.
And that’s what we’re going to do on the right-hand side as well; every time we see an 𝑥, we’re going to replace that by 14. So starting with the first term 𝑥 squared, what does that become when 𝑥 is replaced by 14? All that becomes 14 squared.
The next term is 28𝑥, which remember is a shorthand for 28 times 𝑥, so really there’s a hidden multiplication sign in there; that’s very important. So what happens when we copy that to the line below and change 𝑥 into 14?
Well we get 28 times 14. And of course we’re subtracting that’s from the previous term, 14 squared. And finally the last term on the top line is a seven that we’re adding. Well that doesn’t contain an 𝑥, so what we do? Well we just leave it as it is, so we add seven as above.
Great so now we’ve got a numerical expression, 14 squared minus 28 times 14 plus seven, which we can evaluate either if stick it into a calculator or by hand. So this is the next line of working if we’re doing it by hand. You’ve taken the 14 squared and turned it into 196; 28 times 14 is 392; and the plus seven stays at it is.
And if we evaluate that, we get minus 189. So there we have it. The output of function when the input is 14, which you write like this, is 𝑓 of 14 is equal to minus 189. This works for any function which is given by an algebraic rule as long as you’re careful.
For example, if you take 28𝑥 and substitute in the 14, you might be tempted to just write the 14 next to the 28 as I’ve done here and then read off 2814. This is wrong! You have to remember that 28𝑥 is shorthand for 28 times 𝑥, and so the correct way of substituting as we did in the working beforehand is to write 28 times 14, which we already established was 392.