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).