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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.$f(x)=2 x-5 \quad g(x)=\frac{x+5}{2}$

(a) $x \quad-\infty<x<\infty$(b) $x \quad-\infty<x<\infty$

Algebra

Chapter 1

Functions and their Applications

Section 2

Basic Notions of Functions

Functions

Missouri State University

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

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).

03:18

08:30

For the given functions $f…

10:09

for the given functions $f…

07:26

02:51

Determine (a) $f(x)$ ) and…

04:11

10:25

09:01

06:34

04:01

11:18

For the given functions fa…

04:46

01:22

Find $(a) f \circ g$ and $…

00:35

Let $f(x)=x^{2}$ and $g(x)…

01:35

The functions f and g are …

For this problem, we've been given to functions F of X equals two, X minus five and G FX, which equals X Plus five divided by two. And we want to combine these two functions into two composite functions f of G of X and G of f of X. So what do we mean by a composite function? While composite function like F of G of X is a function where the input toe one function is the output of another, here's another way to think of this. If we just had a function f and let's say I had f of three, that means that everywhere where I have X in my original F function, I'm going to substitute three. Well, in this case, I don't have three that I'm substituting. What I'm substituting is G of X, so whatever, I'll evaluate G of X, and then that is what I'm going to put in to my F function. So how do we actually go? Go about writing this? Well, let's start with F of G F X. My outermost function is F. It's f of G of X, so I'm going to go to my F function which is two times X minus five. What am I putting into that X one of my substituting on substituting G of X so G f X is X plus five divided by two. Now when we simplify this, you can see that my twos they're going to cancel, which gives me X Plus five minus five or just X. So what is the domain of this function when I look at the domain of a composite function after looking to places first, I have to make sure that whatever is in my domain works for my composite function as a whole. When this case is X and X, I could have anything Be X. There's no limits or restrictions on there. I also have to look at my G of X function because number. Evaluate that and then put it into my F function. So I need to go back and look at my GI function to make sure there's nothing that missing from that domain that I would have toe restrict it. And in this case, there's not. I can let this XB anything I want. So here my domain is all real numbers, from negative infinity to positive Infinity. Okay, Now let's look at our other composite function G of f of X. Well, in this case, G is my outside function. So let me go to G. I have X plus five divided by two. What is my ex? One of my substituting in for X in this case on substituting in half of X, which is two X minus five. Well, let's simplify this a bit. In my numerator, I have a negative five in a positive five. The cancel which will love also allow me to cancel the twos in the numerator and denominator. So again, this composite function just gives me X. What about the domain? Well again to places toe Look, I have to look at the composite function as a whole. There are no restrictions to what X could be here. And I have to look at the innermost function f of X. Well, that has doesn't have any restrictions. Either X could be anything I want for my f of x function. So once again, my domain is all real numbers. Negative infinity to positive. Infinity

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