We can think of a function as a transformation. In this lesson, we will look at whether or
not we can reverse that transformation. This will give you another way to build new functions
out of some already-known functions, by taking their inverses.
Definition of inverse functions
The function $\cl"red"{f}$ from a
previous lesson is given in a table below,
along with a new function $\cl"blue"{g}$. They are also graphed on the grids to the left. Notice
that the graphs of $\cl"red"{f}$ and $\cl"blue"{g}$ are the same shape, but with their $x$ and
$y$ coordinates reversed. (This is the same as reflecting the graphs across the dashed line
$y=x$.)
For each $x$ in the table below, compute ${g ∘ f}(x)$.
| Is ${g ∘ f}(x)$ always equal to $x$?
| |
You could also check that ${f ∘ g}(y)$ is always equal to $y$, in the same way. This means
that $\cl"red"{f}$ and $\cl"blue"{g}$ undo each other. That is, when you use $\cl"red"{f}$
to transform a number into another number, $\cl"blue"{g}$ will transform the result back into
the original number, and vice versa.
If $f$ and $g$ are two functions with ${g ∘ f}(x)=x$ for every $x$ in the
domain of $f$, and ${f ∘ g}(y)=y$ for every $y$ in the domain of $g$, then we call $f$ and $g$
inverse functions.
| A new function $\cl"red"h$ is given in the table to the
right, and also graphed on the grid to the left. Is $\cl"red"h$ a one-to-one function?
| |
| | $x$ | $\cl"red"{h(x)}$
|
|---|
| $0$ | $2$
| | $1$ | $3$
| | $2$ | $-4$
| | $3$ | $1$
| | $4$ | $-2$
|
|
If $\cl"red"h$ has an inverse function, we’ll name that function $\cl"blue"{h^{-1}}$ (read as
“$h$-inverse”). Click to show a grid where we can graph
$\cl"blue"{h^{-1}}$.
Because $\cl"red"{h(0)=2}$, we know that
$\cl"blue"{h^{-1}(2)=0}$ (since $\cl"blue"{h^{-1}}$ must reverse the effect of $\cl"red"h$).
Proceeding in this way, fill in the table below for $\cl"blue"{h^{-1}}$. Your answers
will be plotted on the bottom grid to the left.
| $y$ | $\cl"blue"{h^{-1}(y)}$
|
|---|
When a function is one-to-one, we can always find an inverse of it in this way. In fact:
Functions have inverses precisely when they are one-to-one. If $f$ is a
one-to-one function, we will write $f^{-1}$ for the inverse function of $f$.
What are the domain and range of $\cl"red"h$?
What are the domain and range of $\cl"blue"{h^{-1}}$?
If $f$ has an inverse, the domain of $f$ will be the range of $f^{-1}$, and
the range of $f$ will be the domain of $f^{-1}$.
Inverses of some specific functions
Just as you can tell if a relation is a function by looking at its
graph, you can also tell if a function is one-to-one by looking at its graph.
If a function has two or more inputs that give the same
output, it is not one-to-one. Looking
at the two functions shown below, the one on the right isn’t one-to-one because there
are two different inputs that will give you each output greater than 0 (that is, two different
points on the graph at each height greater than zero). The function on the left
is one-to-one because each point on the graph is at a different height.
The function $\cl"red"f$ given by $\cl"red"{f(x)=x^3}$ is graphed on the grid to
the left.
| Based on the graph of $\cl"red"{f(x)}$, is $\cl"red"{f}$ a one-to-one
function?
| |
| Does $\cl"red"{f}$ have an inverse?
| |
| Compute $\cl"red"{f(2)}$.
| |
| Using your previous answer, what is $f^{-1}(8)$?
(Remember that $f^{-1}$ is the name given to the inverse function of
$f$.)
| |
If $\cl"red"{f(x)=x^3}$, then $f^{-1}(y)$ must be the number whose cube is $y$. When $y$ is
positive, we have already defined that number to be $√^{3} y$. As you can see by looking at the
graph, the range of the function $\cl"red"f$ is the set of all real
numbers, so it makes sense to talk about $f^{-1}(y)$ for any real number $y$ (including negative
numbers).
As a result, we will define $√^{3} y$, the cube root of $y$, to be
the value $f^{-1}(y)$, where $\cl"red"{f(x)=x^3}$.
This definition works whether $y$ is negative or positive.
Click to see the graph of $\cl"blue"{f^{-1}(y)=√^{3}y}$
below the graph of $\cl"red"f$. Notice that the two graphs are the same shape, but with the axes
switched.
| What is $\cl"red"{f(-3)}$?
| |
| What is $\cl"blue"{f^{-1}(-27)}$?
| |
| What is $\cl"red"{f(-4)}$?
| |
| What is $\cl"blue"{√^{3}{-64}}$?
| |
The function $\cl"green"f$ given by $\cl"green"{f(x)=x^2}$ is graphed on the grid
to the left.
| Is $\cl"green"{f}$ one-to-one?
| |
| Does $\cl"green"{f}$ have an inverse?
| |
Click to see the graph of a new function $\cl"red"{h}$.
This is a restriction of $\cl"green"{f}$ to a smaller domain: that is,
the domain of $\cl"red"{h}$ is smaller than the domain of $\cl"green"{f}$, but inside the domain
of $\cl"red"{h}$ the functions have the same value at every point.
| Is $\cl"red"{h}$ one-to-one?
| |
| Does $\cl"red"{h}$ have an inverse?
| |
Because $\cl"red"{h(x)=x^2}$ whenever $x≥0$, $h^{-1}(x)$ must be the
non-negative number whose square is $x$. That is, $h^{-1}(x)$ is the square root of $x$.
Click to see the graphs of $\cl"red"{h}$ and
$\cl"blue"{h^{-1}}$.
| Is $\cl"red"{h(x)}$ ever a negative number?
| |
| In the previous question, you saw that we could define the
cube root of a negative number. Is it possible to define the
square root of a negative number in the same way?
| |
If $f$ is a linear function, you can compute $f^{-1}$ using algebra.
For example, the function $\cl"red"f$ given by $\cl"red"{f(x)=2x+1}$ is graphed on the top
grid to the left, and its inverse is graphed on the bottom grid. We’d like to find a formula for
$\cl"blue"{f^{-1}(y)}$. That is, given any $y$ value, our formula should produce the value of
$x=f^{-1}(y)$ such that $f(x)=y$. To find this formula, we start with the equation $f(x)=y$, and
then solve for $x$ in terms of $y$:
$$
\cl"tight"{\table
2x, +1, =, y;
2x, , =, y, -, 1;
x, , =, \colspan 3{1/2(y-1)};
x, , =, {1/2}y, -, 1/2
}
$$
So $$\cl"blue"{f^{-1}(y)={1/2}y - 1/2}$$.
Click to see the graph of $\cl"red"{g(x)=3x-4}$ and its
inverse. Find a formula for $\cl"blue"{x=g^{-1}(y)}$, by solving $g(x)=y$ for $x$ in terms of
$y$.
Now click to see the graph of $\cl"red"{h(x)=x-2}$
and its inverse.
| Using scratch paper or the graph to the left, what is a formula for
$\cl"blue"{h^{-1}}$?
| |
To complete the table below, click the Next buttons to see the
graphs of the linear functions $\cl"red"f$, $\cl"red"g$, and $\cl"red"h$ again, along with
the graphs of their inverses. Enter the slope of each graph, and the formulas you just found for
the inverse functions.
| Function | Slope | Inverse | Slope of
inverse
|
|---|
As you can see from this table:
If $f$ is a linear function with slope $m$ and $m ≠ 0$, then $f^{-1}$ is a
linear function with slope $$1/m$$.
Click to see the graph of the function $\cl"red"k$ given by
$\cl"red"{k(x)=-2x+4}$.
| What is the slope of $k$?
| |
| What is the slope of $k^{-1}$?
| |
| By looking at this graph, find the intercepts of $\cl"red"{k}$.
(The intercepts of a graph are the places where it crosses the $x$-axis and
the $y$-axis.)
| |
Now, click to see the graphs of $\cl"red"{k}$ and
$\cl"blue"{k^{-1}}$.
| Where are the intercepts of $\cl"blue"{k^{-1}}$?
| |
Because $\cl"red"{k}$ and $\cl"blue"{k^{-1}}$ have the same shape except with the coordinates
switched, they also have the same intercepts except with the coordinates switched. This is true
of any linear function:
If $f$ is a linear function with intercepts $(0,b)$ and $(c,0)$, then
$f^{-1}$ is a linear function with intercepts $(b,0)$ and $(0,c)$.