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Now suppose your function is the following set of points:
{ (1, 2), (2, 1), (3, 4), (5, 1) }
The inverse of this function is the same set of points, but the x's and y's have been reversed:
{ (2, 1), (1, 2), (4, 3), (1, 5) }
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This inverse has two points, (1, 2) and (1, 5), that share a common x-value but have different y-values. This means that the inverse is NOT a function.
Graphically, the original function looks like this:
You can find the inverse algebraically, by flipping the x- and y-coordinates, or graphically, by drawing the line y = x...
...and reflecting all the points across it:
It's perfectly okay for the inverse to overwrite the original function's points. The points (2, 1) and (1, 2) of the inverse overwrote the points (1, 2) and (2, 1) of the original function, which is why two red dots went missing when we did the inverse. But this sharing of points between function and inverse is totally okay.
However, you can see from the graph above that the inverse is not a function. This is because there are two points in the inverse that are sharing an x-value. The inverse fails the Vertical Line Test:
There is a quick way to tell, before going to the trouble of finding the inverse, whether the inverse will also be a function. You've seen that you sort of "flip" the original function over the line y = x to get the inverse.
Using this fact, someone noticed that you can also "flip over" the Vertical Line Test to get the Horizontal Line Test.
What is the Horizontal Line Test?
The Horizontal Line Test allows you to check, from the graph of a function, whether that function's inverse will also be a function. If there is any place on the graph of the original function where a horizontal line would cross two more more times, then the inverse of that function will not itself be a function. If all horizontal lines cross at most one spot on the original graph, then the inverse will be a function, too.
As you can see, we can draw a horizontal line through two of the points in the original function (being the original set of points):
Since the original function had two points that shared the same Y-VALUE, then the inverse of the original function will not be a function.
This means, by the way, that no parabola (quadratic function) will have an inverse that is also a function.
In general, if a function's graph does not pass the Horizontal Line Test, then the graphed function's inverse will not itself be a function; if the list of points contains two or more points having the same y-coordinate, then the listing of points for the inverse will have two or more points having the same x-value and thus will not be a function.
So when you're asked, "Will the inverse be a function?" and you're given a graph, draw a horizontal line; if you're given a list of points, compare the y-coordinates.
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Definition of inverse function
Before we start with the inverses of functions, we have to briefly review the functions.
Let’s consider the function $latex f(x)=2x-3$. We know how to evaluate the function f at 2, $latex f(2)=2(2) -3=1$. We can think of the function f as something that transforms 2 into 1 and 4 into 5:
Therefore, since the function f is acting on the numbers and transforming them, we can think of the inverse of f as something that “reverses” the effect of the function f. That is, the inverse of f must take 1 as input and produce 2 and take 5 to produce 4.
Suppose we have the function $latex g(x)=\frac{x+3}{2}$. When evaluating it with input 1, it results in $latex g(1)=\frac{1+3}{2}=2$ and when evaluating it with input 5, it results in $latex g(5)=\frac{5+3}{2}=4$. We can see that function g seems to reverse the effect of function f.
To prove that the function g is the inverse of f, we must show that this is true for any value of x in the domain of f. That is, the function g must take $latex f(x)$ and return x. Then $latex g(f(x))=x$ must be true for all values of x in the domain of f.
One way to verify this is simply by checking if $latex g(f(x))$ returns x:
$latex g(f(x))=\frac{2x-3+3}{2}$
$latex g(f(x))=\frac{2x}{2}$
$latex g(f(x))=x$
Using $latex {{f}^{-1}}$ to denote the inverse of f, we have just shown that $latex g={{f}^{-1}}$.
Graph of inverse functions
We know that the reflection of a point (a, b) with respect to the x-axis is (a, -b) and that the reflection of (a, b) with respect to the y-axis is (-a, b). Now, we want to reflect with respect to the line $latex y=x$.
The following graph illustrates the reflection of the point (a, b) with respect to the line $latex y=x$ to form the point (b, a):
If we have the function $latex f(x)={{x}^3}+1$, then we have $latex f(1)=2$ and the point (1, 2) is on the graph of f. The inverse function of f must take 2 as input and produce 1 as output, that is, $latex {{f}^{- 1}}(2)=1$ and the point (2, 1) is on the graph of $latex {{f}^{-1}}$.
The point (2, 1) is the reflection of the point (1, 2) with respect to the line $latex y=x$. The same happens with the rest of the points in the graph of f, so the inverse is the graph that results when reflecting the graph of f about the line $latex y = x$:
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Some functions do not have inverses. For example, suppose we have the function $latex f(x)={{x}^2}$. The function can take two different numbers and produce the same output, for example, $latex f(3)=9$ and $latex f(-3)=9$. If f had an inverse, this would mean that this function would take 9 to produce both 3 and -3.
However, this runs contrary to the definition of a function, which states that each input should produce only one output. Therefore, there is no function that is the inverse of f.
In terms of graphs, if f had an inverse, then its graph would be a reflection of the graph of f with respect to $latex y=x$:
We can see that the reflected graph does not pass the vertical line test, which means that the graph does not represent a function. We can generalize this as follows:
A function f has an inverse only if when its graph is reflected with respect to y = x, the result is a graph that does pass the vertical line test. But we can simplify this. We can determine before reflecting the graph whether the function has an inverse or not by using the horizontal line test.
Horizontal line test
We have the function f. The horizontal line test tells us that:
- If any horizontal line intersects the graph of f more than once, then f does not have an inverse.
- If any horizontal line does not intersect the graph of f more than once, then f does have an inverse.
Finding inverse functions – Method and examples
Let’s start by considering a simple function $latex f(x)=2x+3$.
The graph of f is a line with slope 2, therefore, it passes the horizontal line test and has an inverse.
There are two steps required to evaluate f at a number x. First, we multiply the x by 2 and then we add 3.
To get the inverse of the function, we must reverse those effects in reverse order. Therefore, to form the inverse function $latex {{f}^{- 1}}$, we start by reversing the sum of 3 by subtracting 3. Then, we reverse the multiplication by 2 by dividing by 2. Then, we have:
$latex {{f}^{- 1}}(x)=\frac{x-3}{2}$
Steps to find the inverse of a function f
Step 1: Replace $latex f(x)$ by y in the equation of the function.
Step 2: Swap the x‘s and y‘s.
Step 3: Solve for y.
Step 4: Replace y by $latex {{f}^{-1}}(x)$.
EXAMPLE 1
Find the inverse of $latex f(x)=5-\frac{x}{3}$.
Step 1: $latex y=5-\frac{x}{3}$
Step 2: $latex x=5-\frac{y}{3}$
Step 3: $latex y=15-3x$
Step 4: $latex {{f}^{{-1}}}(x)=15-3x$
EXAMPLE 2
Find the inverse of $latex f(x)={{x}^3}+5$.
Step 1: $latex y={{x}^3}+5$
Step 2: $latex x={{y}^3}+5$
Step 3: $latex y=\sqrt[3]{{x-5}}$
Step 4: $latex {{f}^{{-1}}}=\sqrt[3]{{x-5}}$
Try solving the following practice problems
Find the inverse of $latex f(x)=3x-4$.
Choose an answer
$latex f^{-1}(x)=\frac{x+3}{4}$
$latex f^{-1}(x)=\frac{x+4}{3}$
$latex f^{-1}(x)=\frac{x-4}{3}$
$latex f^{-1}(x)=3x+4$
Find the inverse of $latex f(x)=\frac{1}{3}x-4$.
Choose an answer
$latex f^{-1}(x)=x+4$
$latex f^{-1}(x)=3x+4$
$latex f^{-1}(x)=3x+12$
$latex f^{-1}(x)=4x+4$
Find the inverse of $latex g(x)={{x}^2}+4$.
Choose an answer
$latex g^{-1}(x)=-{{x}^2}-4$
$latex g^{-1}(x)=\sqrt{x-4}$
$latex g^{-1}(x)=\sqrt{x+4}$
$latex g^{-1}(x)=\sqrt{4x-4}$
See also
Interested in learning more about functions? Take a look at these pages:
- Types of Functions with Graphs
- How to Know If a Function is Symmetric?
- How to Know If a Function is Linear?