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Determine (a) $f(x)+g(x),$ (b) $f(x)-g(x),$ (c) $f(x) g(x)$ (d) $f(x) / g(x)$ when defined.$f(x)=x^{2}+5 \quad g(x)=\frac{x+1}{2}$

(a) $x^{2}+1 / 2 x+11 / 2$(b) $x^{2}-1 / 2 x+9 / 2$(c) $1 / 2\left(x^{3}+x^{2}+5 x+5\right)$(d) $\frac{2 x^{2}+10}{x+1} \quad x \neq-1$

Algebra

Chapter 1

Functions and their Applications

Section 2

Basic Notions of Functions

Functions

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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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02:27

Determine (a) $f(x)+g(x),$…

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03:22

01:11

Find (a) $f(x)+g(x),$ (b) …

05:25

04:27

Find a. $(f \circ g)(x)$ b…

04:44

07:30

Given $f(x)=2 x^{2}+1$ and…

01:36

Find $(f+g)(x),(f-g)(x),$ …

04:06

for this problem, we've been given to functions f of X equals X squared plus five and G of X equals X plus 1/2. And we're going to use our four basic arithmetic operations addition, subtraction, multiplication and division to combine our two functions. And the good news is that these operations work exactly the way you'd expect them the work, even though we're using functions instead of just numbers that we're putting together. For example, let's start with addition ffx plus g of X. Well, we're just going to add the two functions together. F of X is X squared plus five and G of X is X plus 1/2. So we do want to add, like terms. So I haven't x squared, and I'm gonna take this piece and I'm gonna break it apart. This is really X over to plus a half, so I have X squared plus X over to and then I have two constants to put together. So I have 5.5 or 11 halfs. Well, what about subtraction? But we're going to set it up exactly the same way. Just subtracting. Instead of adding I have f of X minus G of X. And again, I'm going to just use that that broken up piece here so I can see the X terminal constant. It makes a little bit easier to put together. Now, don't forget, you do have to distribute that negative sign. So what we really have is X squared plus five minus x over two minus a half. Putting that together, I get X squared minus X over to. And now I have five minus a half, which is nine halfs. Okay, multiplication f of x times G of X. So I have X squared plus five. And I have X plus 1/2. Well, what I'm going to do when I multiply this, I'm going to just have that divided by two here. I'm gonna pull that out is a half in the front because, you know, another way of thinking about this is X plus one times a half. So I'm just gonna pull that half in front, and then I can use binomial expansion or binomial multiplication to multiply those two together. So multiplying that in that I get X cubed plus x squared plus five x plus five. If you want to distribute the one half you can. Otherwise, you could just leave it out in front for me. I think that looks a little bit. It's easier to read for me. So I'm gonna leave it in front and last. We're going to do a quotient F of X divided by G of X. So I have X squared plus five on the top. And the denominator is X plus 1/2. Okay, I don't wanna have that double fraction bar, so I'm going to multiply top and bottom by two, which gives me two X squared plus 10 over X plus one. Now, I do have to be a little careful with this quotient. It's important to know that I can't divide by zero. So I need to check and see what could make that denominator zero. Well, in this case, negative one could So x can't equal negative one. And you do need to make sure you put that on there. We didn't have that anywhere else. We didn't have a next in the denominator anywhere. So this was the only spot that we needed toe kind of that double check for

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