Inverse functions
A mathematical relation $R$R is a mapping of elements in one set to elements in another set. Any number of elements in the first set (usually called the domain) can be mapped to any number of elements in the second set (usually called the range).
Some relations have the property that elements in the domain are mapped to unique elements in the range. Such relations are called functions. As an example the function $y=x^2$y=x2 maps values of $x$x in the domain to unique values of $y$y. We know for instance that $3$3 is mapped to $9$9, and that $4$4 is mapped to $16$16, and that $-3$−3 is mapped to $9$9 and so on. Nowhere would we find a single number in the domain being mapped to two or more different numbers in the range.
Some functions are one-to-one functions. These are functions where each value of $x$x in the domain is mapped to a unique value of y in the range.
A function like $y=2x+3$y=2x+3 is one-to-one because every value of $x$x in the domain is mapped to a different value of $y$y in the range. The function $y=x^2$y=x2 is not one-to-one because, apart from $0$0, there are always two distinct values of $x$x in the domain mapped to each value of $y$y.
If, and only if, a function is one-to one, then a reverse mapping will create what is known as an inverse function.
To explain the idea of a reverse mapping, think about the one-to-one function $y=2x+3$y=2x+3. A reverse mapping is formed by swapping the $x$x and $y$y variables so that $x=2y+3$x=2y+3 and then making $y$y the subject, so that $y=\frac{1}{2}\left(x-3\right)$y=12(x−3).
Note that this new inverse function is still one-to-one as every value of $x$x in the domain is mapped to a different value of $y$y in the range.
Geometrically, swapping the $x$x and $y$y variables around ensures that the function and the inverse function are mirror images across the line $y=x$y=x as shown in the first graph.
In a reverse mapping, the range and domain are reversed as well.
For example, the one-to-one function given by $y=\frac{x-6}{x+4}$y=x−6x+4 has the natural domain as all real numbers excluding $x=-4$x=−4. The range includes all real numbers other than $y=1$y=1. The exclusion of $y=1$y=1 can be seen if we rewrite the function as $y=1-\frac{10}{x+4}$y=1−10x+4.
The inverse mapping is thus given by $x=\frac{y-6}{y+4}$x=y−6y+4 with the domain given by all real numbers other than $x=1$x=1 and the range given by all real numbers other than $y=-4$y=−4.
Since $x=\frac{y-6}{y+4}$x=y−6y+4 we can make $y$y the subject of the inverse function as follows:
| $x$x | $=$= | $\frac{y-6}{y+4}$y−6y+4 |
| $xy+4x$xy+4x | $=$= | $y-6$y−6 |
| $xy-y$xy−y | $=$= | $-4x-6$−4x−6 |
| $y\left(x-1\right)$y(x−1) | $=$= | $-2\left(2x+3\right)$−2(2x+3) |
| $y$y | $=$= | $-2\left(\frac{2x+3}{x-1}\right)$−2(2x+3x−1) |
Here is the graph of the function and its inverse:
Worked Examples
QUESTION 1
Do the following graphs have inverse functions?
- Loading Graph...
A graph of a straight line on a Cartesian plane that passes through points $\left(0,2\right)$(0,2) and $\left(3,3\right)$(3,3). Yes
ANo
B - Loading Graph...
A graph of a parabola that opens upward on a Cartesian plane with its vertex located at $\left(0,2\right)$(0,2). Yes
ANo
B - Loading Graph...
A graph of an exponential function on a Cartesian plane where the curve is asymptotic to the x-axis as x approaches negative infinity. Yes
ANo
B - Loading Graph...
A graph of a cubic function with only one x-intercept, which is $x=0$x=0, on a Cartesian plane. Yes
ANo
B - Loading Graph...
A graph of a cubic function with three x-intercepts, which are $x=-1$x=−1, $x=2$x=2, and $x=4$x=4, on a Cartesian plane. Yes
ANo
B
QUESTION 2
Find the inverse function of $y=x^5+1$y=x5+1.
QUESTION 3
Examine the following graph containing two lines:
Are the lines in the graph inverse functions of each other?
Yes
ANo
B