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

(a) $x^{2}+2 x-3$(b) $-x^{2}+2 x-3$(c) $2 x^{3}-3 x^{2}$(d) $\frac{2x-3}{x^2} (x \neq 0)$

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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for this problem, we have been given to functions F of X equals two X minus three and G of X equals X squared. And we're going to use our four basic arithmetic operations addition, subtraction, multiplication and division to combine our functions. So let's start with addition ffx plus G of X. Well, let's just set this up. F of X is two x minus three plus g of x g of X is X squared. If we had, like terms, we would combine them. But we don't, so we're gonna end up with three terms. Usually it's nice to write them in descending order, although it's not really necessary. You could have left it the way we just had it. Or in descending order. It's X squared plus two X minus three. What if we're going to subtract thes functions while same thing First, we're going to substitute F of X equals two X minus three and G of X equals X squared Again. I would subtract like terms if we had them and distribute that subtraction sign. But I don't have any like terms here. So again, I'm just gonna put these in descending order. Negative X squared plus two X minus three is my subtraction. Okay, let's try multiplication. First F of X. That's two x minus three g of X is X squared. So now we're going to distribute that X squared That gives me two x cubed minus three x squared. So that is my product of my two functions. And finally, we have our division quotient of f of X over g of X, and once again, we're just going to substitute in. That's two X minus three over X squared. Now there's one extra piece we have to do here that we didn't do for the others. But we have a quotient. We now have a denominator. And remember, denominators cannot equal zero because we can't divide by zero, so X squared cannot be zero, which means X can't equal zero. So we do just have to have put that in our final answer that it's important to note that for the quotient, X cannot equal zero

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