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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.$f(x)=\sqrt{2 x+3} \quad g(x)=\frac{x^{2}-3}{2}$

(a) $|x|-\infty<x<\infty$(b) $x \quad x \geq-3 / 2$

Algebra

Chapter 1

Functions and their Applications

Section 2

Basic Notions of Functions

Functions

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Oregon State University

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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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for the given functions $f…

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for this problem, we've been given to functions. F of X equals the square root of two X plus three and G of X equals X squared minus three, divided by two. And we have The goal for this exercise is to create two composite functions f of G of X, n g of F of X. So what exactly is a composite function? Well, let's take a step back and look at a single function. Let's say with the function F if I have f of three, what does that mean? It means I'm going to go to the F function. And everywhere there's an X in the F function, I'm going to substitute in a three. But what if I don't just have a number here instead of a three inside these parentheses? I'm going to have another function G of X. That's a composite function, and we treat it the same way as if it was a number of those parentheses. We're going to go to the F function, and then, instead of ex, I'm going to replace X every instance by G of X. So let's see how that actually works with the functions we've been given f of g of X. So my outer function is f So I'm going to go to my F function, which is the square root of two X plus three. What am I substituting in for X? I'm substituting in G of X and G of X is X squared minus three, divided by two. Okay, let's simplify this a little bit. I have I'm time multiplying by two and dividing by two. So they cancel. So that leaves me with X squared minus three plus three. Well, the minus three and plus three cancel is well, so all I have is the square root of X squared. Well, the square root of something squared is going to be just what I have on the radical X. However, square roots operation on Lee gives us the positive, uh, number. So I have to make this the absolute value of X to show that I'm on Lee ever going to get a positive value back because of the absolute value because of the square root operation. Okay, so that is my composite function. What about the domain? How do we find the domain of a composite function? Well, there's two places we have to look. First we have to look at the innermost function because that G of X is going to be the input into F. So I have to have a valid g of X before I can get to the F part. So let's go look at RG function. Are there any restrictions on the domain here? Well, in this case, no, I could let x be any number I want. I also need toe. Look at the composite function as a whole. Is there any restriction on this domain again? No. I can take the absolute value of any real number. I want to. So with no restrictions, my domain is all real numbers from negative infinity to positive infinity. Okay, what about G of F of X? Well, this time, my outermost function is a G. So I'm going to go to my GI function and start their my GI function is X squared minus three, divided by two. What am I putting in place of X f of X? So my f of x is the square root of two x plus three. Well, now when I simplify, I'm squaring a square root that gives me exactly what was under the radical. And most of this cancels I have a plus three and a minus three. I have a two divided by two. What I have left is just X. Okay? What about the domain for this composite function? Well, like before, we have to check two places first F of X, that innermost function. I have to make sure that I have valid, uh, numbers coming out of that function. So are there any restrictions on the domain there? Actually, yes, there are. With a square root, the piece that's under the square root. In this case, two X plus three has to be non negative, greater than or equal to zero. And if I solve this for X, I get X is greater than or equal to negative three haps. So that's gonna have to be part of my domain. X is greater than or equal to negative three haps. We've checked the innermost function. We also have to check the composite function as a whole. Are there any restrictions on X? No, there aren't. So my domain is all real numbers of X where X is greater than or equal to negative three halves

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