Inverse Function. Graph of the Inverse Function
Let's compare two functions y=f(x) and y=g(x) (see figure). They are both defined on segment [a,b], and their range is segment [c,d]. First function has the following property: for any `y_0` from segment [c,d] there is ONLY ONE value `x_0` from segement [a,b] such that `f(x_0)=y_0`. Geometrically this means that any horizontal lines that intersects y-axis between points c and d, intersects graph of the function y=f(x) only once. Second function doesn't have this property: for example, for value `y_1` line `y=y_1` intersects graph of the function y=g(x) three times. Therefore, in first case for any fixed value `y_0` from segment [c,d] equation `f(x)=y_0` has only one root `x_0`; in second case for some values of y, for example for `y=y_1`, equation `g(x)=y_1` has more than one root.
If function y=f(x) has the following property: for any its value `y_0` equation `f(x)=y_0` has only one root, then we say that function is invertible (or one-to-one).
Definition. Function f is one-to-one if `x_1!=x_2` implies `f(x_1)!=f(x_2)`.
For example. function `y=x^2` is not one-to-one, because `1!=-1`, but `(-1)^2=(1)^2=1`.
Horizontal Line Test: function is invertible if any horizontal line intersects it no more than once.
From figure it can be seen that y=f(x) is invertible, while y=g(x) is not.
If function f is invertible, then expressing x from formula y=f(x) and interchanging x and y will give inverse function. If function is not invertible, then above operation can't be done.
Now, again let's see on the figure. Note, that function y=f(x) is increasing function (and it is invertible), while function y=g(x) is neither increasing, nor decreasing (and it is not invertible).
Fact. If function y=f(x) is defined and increasing (or decreasing) on interval X and its range is interval Y, then it has inverse function, and this inverse function is increasing (decreasing) on interval Y.
Example. Prove that function y=2x-1 has inverse and find it.
Function y=2x-1 is increasing on all number line, therefore, it has inverse function. To find this inverse function, express x in terms of y: `y+1=2x` or `x=1/2(y+1)`.
Now interchange x and y: `y=1/2(x+1)`. This is inverse function.
If point (x;y) belongs to the graph of the function y=f(x), then point (y;x) belongs to the graph of the inverse function. That's why graph of the inverse function is symmetric about line y=x to the graph of the function y=f(x).
So, to draw graph of the inverse function to y=f(x), draw function y=f(x) and then draw inverse symmetrically about line y=x.
Foe example, if `y=x^n`, where `x>=0`, n is natural number, n>1 then `x=root(n)(y)`. Interchanging x and y gives that `y=root(n)(x)`. Graphs of two mutually inverse functions `y=x^n` and `y=root(n)(x)` are symmetric about line y=x (see figure).