Functions

[WTF?]

Hello, I'm in need of another push again...

It says:

Let \(\displaystyle g(x)=x^2\)

a.\(\displaystyle g(x+h)\)
b.\(\displaystyle g(x+h) - g(x)\)
c.\(\displaystyle (g(x+h) - g(x))/(h)\)

Okay, I'm stuck on all...I'm not entirely sure how to approach this, but would I need to substitute \(\displaystyle x^2\)? It seems kind of obvious and delicate....

Thanks again for any replies.

 

[soroban]

Hello, WTF?!

"Substitute \(\displaystyle x^2\)" ? . . . You're forgetting the basics . . .

Let \(\displaystyle g(x)\,=\,x^2\)

a) Find: \(\displaystyle g(x+h)\)

Okay, some baby-talk . . .

What does \(\displaystyle g(4)\) mean?
\(\displaystyle \;\;\)It means "Replace \(\displaystyle x\) with \(\displaystyle 4\)".

So we have: \(\displaystyle \,g(4)\,=\,4^2\,=\,16\)

So what does \(\displaystyle g(x+h)\) mean?
\(\displaystyle \;\;\)It means "Replace \(\displaystyle x\) with \(\displaystyle x+h\)".

So we have: \(\displaystyle \.g(x+h)\:=\x\,+\,h)^2\:=\:x^2\,+\,2xh\,+\,h^2\)

b) Find: \(\displaystyle g(x+h)\,-\,g(x)\)

What does it say?
\(\displaystyle \;\;\)It says: Find \(\displaystyle g(x+h)\), then subtract \(\displaystyle g(x)\).

Well, we just found \(\displaystyle g(x+h)\) and we know what \(\displaystyle g(x)\) is.
\(\displaystyle \;\;\)So we have: \(\displaystyle \,g(x+h)\,-\,g(x)\;=\;(x^2\,+\,2xh\,+\,h^2)\,-\, x^2\;=\;2xh \,+\,h^2\)

c) Find: \(\displaystyle \L\frac{g(x+h)\,-\,g(x)}{h}\)

What does it say?

It says: Take what we found in part (b), and divide by \(\displaystyle h\).

So we have: \(\displaystyle \L\,\frac{2xh\,+\,h^2}{h}\)\(\displaystyle \;=\;\L\frac{\not{h}(2x\,+\,h)}{\not{h}}\)\(\displaystyle \;=\;2x\,+\,h\)