[tex]\text{A function is a relationship or expression that involves}[/tex] [tex]\text{one or more variables.}[/tex]
[tex]\text{When we are given Composed Functions), we have the ability}[/tex] [tex]\text{to combine them in such a way that the}[/tex] [tex]\text{outcomes of one function becomes the other}[/tex]
[tex]\text{For example - }[/tex]
If [tex]f(x)=3x-1[/tex] and [tex]g(x)=x^3+2[/tex], then what is[tex]f(g(3))[/tex]?
[tex]g(x)=x^3+2[/tex]
[tex]g(x)=(3)^3+2[/tex]
[tex]= 29 [/tex]
[tex]\text{Since}[/tex] [tex]g(3) = 29[/tex] then [tex]f(g(3))=f(29)[/tex]
[tex]\text{Now let's evaluate}[/tex] [tex]f=(29)[/tex]
[tex]f(x) = 3x-1[/tex]
[tex]f(29)=3(29)-1[/tex]
[tex]=86[/tex]
[tex]f(g(3))=f(29)=86[/tex]
[tex]\text{To find the value of g(x) we need to substitute}[/tex] [tex]\text{a number into the function's formula: }[/tex]
[tex]g\left(x\right)=6\left(4\right)x[/tex]
[tex] \dfrac{d}{dx} (6*4x)[/tex]
[tex]6*4\dfrac{d}{dx}(x)[/tex]
[tex]6*4*1[/tex]
[tex]= 24[/tex]
[tex]\text{The "rate of change" is the slope of a function .}[/tex]
[tex]\text{Formula:}[/tex] [tex] \dfrac{\text{(change in f(x) }}{\text{(change in 'x')}} [/tex]
[tex]\text{In Section A:}[/tex]
[tex]\text{Length of section}[/tex] = (1 - 0) = 1
[tex]f(1) = 6(4)x[/tex]
[tex]f(0) = 0[/tex]
[tex]\text{Change in the value of the given function}[/tex] = [tex](24 - 0) = 24[/tex]
[tex] \dfrac{\text{ (change in the value of the function)}}{\text{(size of the section)}} [/tex] [tex] \dfrac{24}{1} = 24 [/tex]
