It is often required to restrict the natural domain of certain many-to-one functions in order to construct a related one-to-one function that preserves the range of the original function.
For example, consider the simple parabola given by the function $f\left(x\right)=x^2$f(x)=x2. Note that $f\left(-x\right)=f\left(x\right)$f(−x)=f(x) for all $x$x in the natural domain. Hence the function is many-to-one.
One way to construct a related function without disturbing the range of the function is to exclude the left arm of the parabola as shown in this graph. Note that the right arm could just as easily been excluded.
This new function is given by $f(x)=x^2,x\in[0,\infty)$f(x)=x2,x∈[0,∞) so that $x=0$x=0 is included in the restricted domain. Note that the range of the function is unaltered with this restriction.
Rationale for the restriction
The inverse of a function which is not one-to-one is not a function itself. A many-to-one function has an inverse which is one-to-many, and so the new domain elements of the inverse relation are not mapped to unique range values. Thus the inverse cannot be a function.
To avoid this problem a suitable restriction on the domain of a many-to-one function can turn it into a one-to-one function without disturbing the range. Then the inverse of this restricted function is guaranteed to be a one-to-one function.
For example, to derive the inverse for the restricted function $y=x^2,x\in[0,\infty)$y=x2,x∈[0,∞) shown above, we simply swap the variables and exchange the domain and range expressions.
Thus we can write:
| $R:$R: $y$y | $=$= | $x^2$x2, $x\in[0,\infty)$x∈[0,∞), $y\in[0,\infty)$y∈[0,∞) |
| $R^{-1}:$R−1: $x$x | $=$= | $y^2$y2, $y\in[0,\infty)$y∈[0,∞), $x\in[0,\infty)$x∈[0,∞) |
| $\therefore y$∴y | $=$= | $\sqrt{x}$√x, $y\in[0,\infty)$y∈[0,∞), $x\in[0,\infty)$x∈[0,∞) |
From the graph of the restricted function and its inverse, you should be able to see that both are one-to-one functions:
Where to make the restriction becomes obvious from the graph of the unrestricted function.
For example, the domain of the semicircle $f\left(x\right)=\sqrt{r^2-x^2},-r\le x\le r$f(x)=√r2−x2,−r≤x≤r needs to be restricted to $0\le x\le r$0≤x≤r.
The function $y=\left|x\right|$y=|x| where $x\in\left(-\infty,\infty\right)$x∈(−∞,∞) needs a restriction on the domain given by $x\in[0,\infty)$x∈[0,∞).
In general, the best way to find the restriction required to ensure that the inverse is one-to-one is to sketch the unrestricted function first. Then, making sure the range is undisturbed, decide on a suitable restriction so that the function, and consequently its inverse, become one-to-one.
Worked Examples
QUESTION 1
State a suitable restricted domain for $f\left(x\right)=\left(x+3\right)^2$f(x)=(x+3)2 so that the function is one-to-one and the range remains the same.
QUESTION 2
State a suitable restricted domain for $f\left(x\right)=-\sqrt{x^2-4}$f(x)=−√x2−4 so that the function is one-to-one and the range remains the same.
QUESTION 3
Find the inverse $y$y of the one-to-one function $f\left(x\right)=\left|x+4\right|$f(x)=|x+4|, which has a restricted domain of $x$x$>=$>=$-4$−4.