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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.$f(x)=3 x+7 \quad g(x)=2 x^{2}-3$

(a) $2 x^{2}+3 x+4$(b) $-2 x^{2}+3 x+10$(c) $6 x^{3}+14 x^{2}-9 x-21$(d) $\frac{3x+7}{2x^2-3} (x \neq \pm \sqrt{\frac{3}{2}})$

Algebra

Chapter 1

Functions and their Applications

Section 2

Basic Notions of Functions

Functions

Campbell University

McMaster University

Baylor University

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).

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for this problem, we've been given to functions. F of X equals three x plus seven and G X equals two X squared minus three. We're going to use the four basic arithmetic operations addition, subtraction, multiplication and division thio to combine these two functions. F N g. No, just because we're doing functions and not numbers. Thes arithmetic operations still work in exactly the same way that you would expect them thio. For example, if we add ffx plus G of X, which is going to add the values of these functions, F of X is three x plus seven g of X is two x squared minus three toe Add These were just going to add are like terms. So that gives us two X squared plus three x plus four. Hey, with subtraction f of X minus G of X. We're just going to subtract the values of these functions. That gives us three X plus seven minus two X squared minus three. Now, don't forget that you have Thio distribute that negative science. What we really have is three x plus seven minus two x squared, plus three. So again, we're going to combine our like terms. I like to put this in descending order. So negative two X squared plus three X plus 10. Okay, let's try multiplication F of X times G F X. Well, that's going to give us three X plus seven times two X squared, minus three. So to buy no meals, we're just going to have to expand these, um, using foil. We're gonna multiply this out. So our firsts give us six X cubed outer negative nine x inner plus 14 x squared and last minus 21. You can leave the answer just like this. There are no common factors to put together. No like terms. However, a lot of people do like to have it in descending order, So I'm just going to write it again and change the order. So I have descending powers of X. Okay, Last one f of X, divided by G of X Well, f of X is my numerator three x plus seven. My denominator is two x squared minus three. Now, this piece, so has one extra caveat that we have to put in. We can't let the denominator equal zero, so I might have some numbers that X can't equal, so we need to take a look at our denominator, set it equal to zero and see if there's any values that we have to exclude. So this would give us two X squared equals three or X squared equals three halfs. So taking the square root X cannot be plus or minus the square root of three over to remember to grab both the plus and the minus there. So two numbers to exclude because I can't divide by zero. Otherwise, that is our function off F of X, divided by G F X.

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