# Grand Canyon University Logarithms & Identity Functions Questions

Provide screenshots or explanations of how you solved the problems.

1. This DQ is dealing with the composition operator of functions. That is, performing one function and then immediately using that output as the input to another possible different function. Here is my problem:

f(x) = 4x + 3

g(x) = sqrt(4 – x2)

to compute (f o g)(x) (which is read as “f of g of x”, the circle in the middle is the composition operator (like a + sign but for composition)). I would write:

f(g(x)) = f( sqrt(4 – x2)) = 4( sqrt(4 – x2)) + 3

then there is no more algebra to simplify.

for (g o f)(x) which is the other way around we have:

g(f(x)) = g(4x+3) = sqrt(4-(4x+3)2) = sqrt(4 – 16×2 – 24x – 9) = sqrt(-5- 16×2 – 24x)

then there is no more algebra to simplify.

Some things to discuss possibly:

1. If a function f composed with itself gives the identity function (i.e f(f(x)) = x) then what can we say about f?

2. On an unrelated note, do you think mathematics is discovered or invented? Is there a difference between the words?

NThis DQ switches over to logarithms: https://en.wikipedia.org/wiki/Logarithm which is NOT an algorithm: https://en.wikipedia.org/wiki/Algorithm

Logarithm is the quintessential slow growing function because it is the inverse (“opposite of”) the exponential function which is the quintessential fast functio

The notation is this: y = logb(x) means that by = x. So it is finding which value of “y” to put as the exponent onto the base “b” to get a value of “x” out

Easy one: log2(4) = 2 because 22 =

Another easy one: log5(25) = 2 because 52 =

Hard one: log6(35) = ?? we need a calculator for this one because theres no whole number to put as a power of 6 to get 35. It is almost 2 right because 62 = 36 and 36 is almost 35. It turns out that log6(35) = 1.9843 using a calculat

Here are some interesting real world applications: https://betterexplained.com/articles/using-logs-in-the-real-world

I can tell you for sure that logs are used every day in my line of work (finance and insurance) mainly in regards to finding unknown interest rates given other criteria about loans/bonds/etc

some discussion point

1. why would the logathim at 0 not have a defined value

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