If h(x)=f(x) g(x) and f(x)=2 x+5 determine g(x) a) h(x)=6 x+15 b) h(x)=-2 x^2-5 x c) h(x)=2 x √(

If h(x)=f(x) g(x) and f(x)=2 x+5 determine g(x) a) h(x)=6...

Key Concepts

Function Multiplication

This concept involves understanding how two functions combine when multiplied together. In problems where one function is represented as a product of two other functions, knowing the properties of function multiplication is crucial for decomposing or recomposing the expressions involved.

Algebraic Manipulation

Algebraic manipulation is key when isolating an unknown function from an equation. Techniques such as factoring and rearranging terms allow one to express an unknown function in terms of known functions or expressions, which is essential for solving equations derived from function relationships.

Polynomial Division

When the functions involved are polynomial expressions, polynomial division becomes an important tool. It involves dividing one polynomial by another to determine an unknown factor or quotient. This method is particularly useful in extracting a function from a factored product when one of the factors is already known.

Transcript

00:01 All right, in this problem, we're given a statement of h of x is equal to f of x times g of x.

00:05 They tell me that f of x is 2x plus five.

00:07 And then for all the problems, they're giving, they're giving me the h of x and what to find g of x.

00:11 So what i did before actually did any of the problems is i kind of just sit in a very general way.

00:17 So if i already know what h of x is already, say that's given to me, i know my f of x is equal to 2x plus 5.

00:24 Remember i'm solving for g of x.

00:26 What would i have to do? so essentially i would just divide everything by 2x plus.

00:30 5.

00:35 So for all these problems right here, just to kind of make the process a little bit faster, we're going to take whatever the h of x that we're given and divided by 2x plus 5.

00:43 Okay.

00:44 And just keep in mind automatically, well, no, for a couple of them, it may not, the main may not happy factor in.

00:52 All right.

00:52 So if you look at a right here, we were given h of x being 6x plus 15.

01:00 So again, since i already did a lot of the work right in the beginning, i'm just literally taking what it says.

01:07 So i'm just doing 6x plus 15 and dividing dividing by 2x plus 5.

01:12 Now, it's really nothing else i can do that in terms of simplification, but i can just state that x -canon equal negative 5 halves.

01:21 So again, it was easier for me to do all that work in advance, so i don't have to redo it again.

01:26 All right, the next one, h of x, a different function.

01:29 We got negative 2x squared minus 5x.

01:33 And again, to get to g of x right here, i'm just simply going to take this negative 2x squared minus 5x and divide by 2x plus 5.

01:41 For this one right here, what i can do is factor out of negative x...

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