In Exercises 97-98, find f(-x)-f(x) for the given function f Then simplify the expression. f(x)=

step 1 Okay, so we are on question 97. So we have F of X is going to be equal to X to the third plus X

In Exercises 97-98, find f(-x)-f(x) for the given function f...

Key Concepts

Function Evaluation and Substitution

This concept involves replacing the variable in the function with a specific value or expression. In the context of the problem, it entails substituting -x for x in the function to obtain f(-x) and then using this result to compute f(-x) - f(x). It is a fundamental technique in algebra that enables us to examine how a function behaves when its input changes.

Simplification of Algebraic Expressions

Once the substitution has been made, the next key step involves simplifying the resulting algebraic expressions. This process includes expanding terms, combining like terms, and canceling out opposing terms where applicable. Simplification is crucial to reduce the expression to its most elementary form, making further analysis or conclusion easier.

Function Symmetry

Exploring f(-x) in relation to f(x) relates to the concept of symmetry in functions, particularly even and odd functions. An even function satisfies f(-x) = f(x), while an odd function satisfies f(-x) = -f(x). Although the primary objective here is to compute f(-x) - f(x), understanding symmetry provides insight into the behavior of the function under a sign change of the input, which can help in further analyses of the function's properties.

Transcript

00:01 Okay, so we are on question 97.

00:08 So we have f of x is going to be equal to x to the third plus x minus 5.

00:20 And we want to figure out f of negative x minus f of negative x minus f of x.

00:29 So let's see, we are going to do, let's do the first part first.

00:34 So what's f of negative x? f of negative x means we plug in negative x whenever we see x in the original equation, so right over here basically.

00:46 So let's do that.

00:47 F of negative x.

00:48 So let's plug a negative x for that.

00:50 So we're going to have negative x to the third plus negative x minus 5.

00:56 So it's going to be negative x times negative x times negative x.

00:59 So it's going to be, let me draw it out for you guys actually.

01:02 So it'll be easier to understand times negative x.

01:06 Plus minus x so that's just minus x then minus 5 and then we're going to do so negative x and negative x is positive x squared times negative x so negative x to the third minus x minus 5 so that's our answer for an f to the negative x now f of x is going to be well f x is just f x right so we're just going to have x right so we're just going to have x to the third plus x minus 5 so now we have to subtract it to we have to do negative x to the third minus x minus 5 minus x to the third plus x minus 5 so now let's retract it to remember we have negative minus all of this stuff so we can just distribute the negative to everything individually so we can do negative x to the third minus x minus 5 then minus x to the third minus x and minus minus plus see so i just distributed this to every one of them.

02:10 So you can think it as like minus one almost, plus minus one to all the stuff...

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