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Determine (a) $f(x)$ ) and the domain of the composite function, (b) $g(f(x))$ and the domain of the composite function.$f(x)=x^{2}+3 \quad g(x)=\sqrt{x-1}$

(a) $x+2 (x \geq 1)$(b) $\sqrt{x^{2}+2 } (-\infty<x<\infty)$

Algebra

Chapter 1

Functions and their Applications

Section 2

Basic Notions of Functions

Functions

Missouri State University

Campbell University

McMaster University

Lectures

01:43

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).

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00:31

for this problem, we have been given to functions. F of X equals X squared plus three and G of X equals the square root of X minus one. And for this problem we want to find are two composite functions f of G of X and G of F of X, and we're going to verify their domains while we're creating these. Now let's take a step back and see what we mean with a composite function. Typically, let's say I have just a single function f of X. If I do something like f of three, it means I'm going to the F function. And then everywhere where there is an ex, I'm going to substitute in a three. Well, if I have a composite function f of G f X, it means the same thing. I'm still gonna be going to the F function that outermost functions where I look first, So I'm going to the F function. What am I putting in place of X? I'm putting g of X in place of X, so it's not just putting in a single number, it's gonna be putting in probably some sort of expression. So let's take a look with our given examples to see what that looks like. I have f is my outermost function. So let's go to my f function. The F function says I have X squared plus three. So that is X squared plus three. I'm giving myself a nice being place. Um, Gap for the X because what I'm gonna do is I'm going to replace the X with what's inside these parentheses here, which is g of X g of X, is the square root of X minus one. So let's simplify. I'm squaring a square root that gives me what's under the radical and I combine that gives me X Plus two. So what about my domain? Well, X Plus two has nothing excluded from its domain, But we have to do not only my final function, but G of X is well because G of X is that inside piece. So technically, we're evaluating G of X first, then putting it into our F function. So my final composite function things have to be in the domain not only of the composite function, but of G of X, because that's my input into the function f now G of X is a square root, which means what's under the square root X minus one has to be greater than or equal to zero, so X has to be greater than or equal toe one. So there's my domain. My domain is all X numbers where X is greater than or equal toe one. Okay, let's look at our other composite function G of F of X. Well, in this case, G is my outside function, So I'm going to go find my G function, which says it's the square root of X minus one. What is X one of my substituting in I'm substituting F of X and F of X is X squared plus three. And if I simplify that under that radical, I end up with a square root of X squared plus two. So let's take a look at our domain. We know we have Thio satisfy the domain of my input function, which is F of X. That's the one that's going into my G function. F of X is X squared plus three. That domain is all real numbers. There's nothing excluded there. Now. My final uh, composite function does have a square root, So I need to make sure that X squared plus two is greater than or equal to zero. Well, it always will be. X squared is always zero or bigger, I add two. That's definitely bigger than zero. So every X will satisfy this. So my domain is all riel numbers. Another way we can write, that is it's from negative infinity to positive infinity.

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