Question 98271
1)

To find average rate between two points on a graph, simply draw a line through those two points and find the slope 


So the x values of the points are given as {{{x=-3}}} and {{{x=3}}}


Now we need to find the y values at the given x-values:


Plug in {{{x=-3}}} to find the value of y at {{{x=-3}}}


{{{f(-3)= (-3)^2 + 10(-3) +16}}} Plug in {{{x=-3}}}



{{{f(-3)= -5}}} Evaluate




Now plug in {{{x=3}}} to find the value of y at {{{x=3}}}


{{{f(-3)= (3)^2 + 10(3) +16}}} Plug in {{{x=3}}}



{{{f(-3)= 55}}} Evaluate


Now that we have the two points (-3,-5) and (3,55) we can find the slope through these points


{{{m=(y[2]-y[1])/(x[2]-x[1])}}} Now start with the slope formula



{{{m=(55--5)/(3--3)}}} plug in {{{x[1]=-3}}}, {{{x[2]=3}}}, {{{y[1]=-5}}} and {{{y[2]=55}}}



{{{m=(60)/(6)}}} Subtract



{{{m=10}}} Divide



So the average rate of change is 10



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2)

{{{K(v)=(2v^2-9)/(v^2-v-30)}}}Start with the given function 



{{{v^2-v-30=0}}} Set the denominator equal to zero





{{{(v-6)(v+5)=0}}} Factor the left side (note: if you need help with factoring, check out this <a href=http://www.algebra.com/algebra/homework/playground/change-this-name4450.solver>solver</a>)




Now set each factor equal to zero:


{{{v-6=0}}} or {{{v+5=0}}}


{{{v=6}}} or {{{v=-5}}}  Now solve for v in each case





Since {{{v=6}}} or {{{v=-5}}} make the denominator equal to zero, that means we must exclude these values from the domain.


So our domain is: v is the set of all real numbers except {{{v<>6}}} or {{{v<>-5}}}


Which looks like this in interval notation:

*[Tex \Large \left(-\infty, -5\right)\cup\left(-5, 6\right)\cup\left(6,\infty \right)]


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3)

{{{f(x)=(1)/(x-20)}}}Start with the given function 



{{{x-20=0}}} Set the denominator equal to zero



{{{x=20}}} Add 20 to both sides



Since {{{x=20}}} makes the denominator equal to zero, that means we must exclude this value from the domain.


So our domain is: x is the set of all real numbers except {{{v<>20}}}


Which looks like this in interval notation:

*[Tex \Large \left(-\infty, 20\right)\cup\left(20,\infty \right)]