Still can't figure it out? Just email us to get a detailed solution to your problem before midnight at:
Quickmathnotes@gmail.com
The inverse of a function \(f(x)\) is written as \(f^{-1}(x)\). Basically, it does the opposite of \(f(x)\), so if \(f(x)\) divides, \(f^{-1}(x)\) multiplies, etc. etc. This means that \(x\) in \(f(x)\) is \(y\) in \(f^{-1}(x)\). There are a few properties of inverse functions that you should be familiar with.
1. if \(f(a) = b\), then \(f^{-1}(b)=a\)
2.\(f(f^{-1}(x)) = x\)
3. The inverse of \(f^{-1}(x)\) is \(f(x)\)
Also, note that to find the inverse of a function, all we need to do is replace \(f(X)\) with \(y\), and then change every \(y\) to an \(x\) and every \(x\) to a \(y\)
We will cover linear functions in example one, cubics in example two, and rational functions in example three.
Find \(f^{-1}\) of \(f(x) = 5x-3\)
First, we replace \(f(x)\) with y, to get: $$y = 5x-3$$ Next, we turn \(y\) into \(x\) and \(x\) into \(y\) to get: $$x = 5y-3$$ Next, we solve for \(y\) to get: $$y= \frac{x}{5} + \frac{3}{5}$$ Therefor: $$f^{-1}(x) = \frac{x}{5} + \frac{3}{5}$$
In this example we will take the inverse of a cubic function
Find the inverse of \(f(x) = 2x^3 -7\)
First, we replace \(f(x)\) with y, to get: $$y = 2x^3-7$$ Next, we turn \(y\) into \(x\) and \(x\) into \(y\) to get: $$x =2y^3 -7$$ Next, we solve for \(y\) to get: $$y= \sqrt[3]{\frac{x}{2}+\frac{7}{2}}$$ Therefor: $$f^{-1}(x) = \sqrt[3]{\frac{x}{2}+\frac{7}{2}}$$
Finally we will take the inverse of a rational function. The algebra is slightly harder here, but the basic process is exactly the same
If \(f(x) = \frac{4x-1}{x+2}\)
First, we replace \(f(x)\) with y, to get: $$y=\frac{4x-1}{x+2}$$ Next, we turn \(y\) into \(x\) and \(x\) into \(y\) to get: $$x =\frac{4y-1}{y+2}$$ Next, we multiple both sides by \(y+2\) to get: $$x(y+2)=4y-1$$ Then we solve for \(y\) $$xy+2x = 4y-1$$ $$xy-4y = -1-2x$$ $$y(4-x) = 2x+1$$ $$y= \frac{2x+1}{4-x}$$
Conlcusion:
You can now find the inverse of every type of function using this method.