Given functions f and g, (b)(g∘ƒ)(x) and its domain. See Examples...

Question

Given functions f and g, (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7. ƒ(x)=x^3, g(x)=x^2+3x-1

Accepted Answer

Hello, everyone. We've been asked to determine G of F of X for the functions F of X equals X to the third and G of X equals to X squared minus eight X plus nine. Also, we need to find the domain of G of F of X. I'm going to rewrite G F F of X with parentheses. So that means that my F of X function is going inside the parentheses for my G, I write it this way, that way I can see which thing I'm plugging in where. So this is like saying, okay, I want G of F of X. So F of X is going in the parentheses and since F of X was X to the third, this is like saying, I want to do G of X to the third. So this means I'm gonna plug X to the third in four X in the G function. So that will look like two times X to the third, raise the second power -8 times x to the third plus nine. So I'm going to clean this up a little bit. I recall that when I raise the power to a power, I multiply the exponents. So the two stays and then X to the third to the second, power is X to the sixth. I don't need to keep parentheses between my minus eight and my X to the third. So I'll write negative eight X to the third and then plus nine looking at it quickly, I want to see is there anything else I can simplify or that could be combined? And there is not. So G of F of X is going to equal to X to the sixth minus eight, X to the third plus nine. In this example, this is part of our solution. But this is not the full answer as we have yet to find the domain. So recalling that a domain is the X values that will make it. So the function results in a real number, I asked myself, is there any number I can plug in positive or negative that will not result in a real number? And I can't think of one. Everything I plug in will come out with a number. So our domain is all real numbers because it doesn't matter which real number I plug in. There's nothing stopping it from being another real number. So now that we have both parts of our answer, looking at my answer choices, it is answer choice D have a nice day.

Given functions f and g, (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7. ƒ(x)=x^3, g(x)=x^2+3x-1

Key Concepts

Function Composition

Domain of a Function

Polynomial Functions