Given that the function 𝑓 𝑥 equals 19𝑥 squared and the function 𝑔 𝑥 equals negative two 𝑥, determine 𝑔 circle 𝑓 of 𝑥 in its simplest form and evaluate 𝑔 circle 𝑓 of one.
The first thing we notice about this question is this notation, and I called it 𝑔 circle 𝑓. It can also be known as 𝑔 composite 𝑓. And 𝑔 composite 𝑓, well it helps to understand what it is cause it’s actually a composite function, so what does 𝑔 circle 𝑓 or 𝑔 composite 𝑓 actually mean?
Well 𝑔 circle 𝑓 is equal to 𝑔𝑓𝑥, and what this means is first you apply 𝑓 and then you apply 𝑔 to the result. So I can actually think of it in terms of a function machine, cause there we have our input, and then we’re gonna complete function 𝑓, which is gonna be 19 multiplied by our input all squared, and then we’re gonna take the value we get from that; that becomes our new input, and then we’re gonna complete our function 𝑔, which is negative two multiplied by that input, and then that will give us our final output.
So as you can see, 𝑔 circle 𝑓, we now know what it is and we now know how to use it, so we’re gonna use it to solve this problem. So we have our two functions and actually if we think about it, our input is 𝑥, so if we complete the first function, so function 𝑓 on 𝑥, we’re gonna get 19𝑥 squared.
So now what we need to do is, to solve this and find out what 𝑔 circle 𝑓 is, is we’re gonna actually substitute in 19𝑥 squared for 𝑥 in our function of 𝑔𝑥. So we can say that 𝑔 circle 𝑓 is equal to negative two multiplied by 19𝑥 squared, which gives the final answer of 𝑔 circle 𝑓 is equal to negative 38𝑥 squared.
Okay and we can check that using our function machine. So this here, our input, is 𝑥 and then it’ll go through our first stage of our function machine, which would be 19 multiplied by 𝑥 squared, which gives us 19𝑥 squared. And then we complete the final part, which would be 𝑔, so we’ve got a new input that is negative two multiplied by the new input of 19𝑥 squared, which gives us negative 38𝑥 squared. Fantastic! This is what we need.
And so we’ve now found 𝑔 circle 𝑓. Having now found 𝑔 circle 𝑓, we can actually now solve the second part of our problem, which is evaluating 𝑔 circle 𝑓 one. What this actually means is that we want to substitute 𝑥 equals one, so we want to make the value of 𝑥 equal to one in our composite function.
So this is gonna give us negative 38 multiplied by one squared, which gives us our answer of 𝑔 circle 𝑓 or 𝑔 composite 𝑓 of one, where 𝑥 is equal to one, is equal to negative 38.
So there we have it. We’ve solved both parts of the problem. So the key thing to remember from this question is that 𝑔 circle 𝑓 or 𝑔 composite 𝑓 means that we’re gonna do the 𝑓-function first and then the 𝑔-function upon it, and it’s the same as 𝑔 of 𝑓 of 𝑥.