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So you are asked to find the inverse of a function f(x).
The inverse function is denoted by f -1(x).
To help with this we can use the identity f(f -1(x))=x.
Now, we need to define y=f -1(x).
x=f(f -1(x))=f(y)=2y+1 As f(y) is similar to f(x) but with the variable change of x to y
Hence, we need to solve:
x-1=2y Minus 1 from each side of the equation
½(x-1)=y=f -1(x) As we defined f -1(x)=y
Therefore, we have found the inverse function: f -1(x) = ½(x-1)
We can continue further and find the domain and range of an inverse function using the identities:
Domain f(x) = Range f -1(x)
Range f(x) = Domain f -1(x)
Integration by parts formula: ∫ u*dv/dx = uv - ∫ du/dx*v dx
To solve this problem we need to use a trick by thinking of lnx as lnx*1
So we can choose: u=lnx, dv/dx=1
The next step is to find du/dx and v.
du/dx=1/x As we have differentiated each side with respect to x
v=x By integrating each side with respect to x
Now we have all the required parts to use the integration by parts formula.
∫ lnx = lnx*x – ∫ 1/x*x dx
= xlnx – ∫ 1 dx
= xlnx – x + c
An arithmetic sequence has a constant difference between each term.
For example: 2,4,6,8,10,12,…
We can see clearly that all the terms differ by +2.
We call this the common difference, d.
A geometric sequence has a constant ratio (multiplier) between each term.
An example is: 2,4,8,16,32,…
So to find the next term in the sequence we would multiply the previous term by 2.
This is called the common ratio, r.
These sequences are closely related as they both have the same first term, but I hope you can see how different they become if they have a common difference or a common ratio.
We can create a decreasing arithmetic sequence by choosing a negative common difference.
Similarly, a decreasing geometric sequence would have a common ratio of less than 1.