How to determine whether a function is inverse or not. Inverse functions - definition and properties

Let the sets $X$ and $Y$ be included in the set of real numbers. Let's introduce the concept of an invertible function.

Definition 1

A function $f:X\to Y$ mapping a set $X$ to a set $Y$ is called invertible if for any elements $x_1,x_2\in X$, from the fact that $x_1\ne x_2$ it follows that $f(x_1 )\ne f(x_2)$.

Now we can introduce the concept of an inverse function.

Definition 2

Let the function $f:X\to Y$ mapping the set $X$ into the set $Y$ be invertible. Then the function $f^(-1):Y\to X$ mapping the set $Y$ into the set $X$ defined by the condition $f^(-1)\left(y\right)=x$ is called the inverse for $f( x)$.

Let us formulate the theorem:

Theorem 1

Let the function $y=f(x)$ be defined, monotonically increasing (decreasing) and continuous in some interval $X$. Then in the corresponding interval $Y$ of values ​​of this function it has an inverse function, which also monotonically increases (decreases) and is continuous on the interval $Y$.

Let us now introduce directly the concept of mutually inverse functions.

Definition 3

Within the framework of Definition 2, the functions $f(x)$ and $f^(-1)\left(y\right)$ are called mutually inverse functions.

Properties of mutually inverse functions

Let the functions $y=f(x)$ and $x=g(y)$ be mutually inverse, then

$y=f(g\left(y\right))$ and $x=g(f(x))$

The domain of definition of the function $y=f(x)$ is equal to the domain of value of the function $\ x=g(y)$. And the domain of definition of the function $x=g(y)$ is equal to the domain of value of the function $\ y=f(x)$.

The graphs of the functions $y=f(x)$ and $x=g(y)$ are symmetrical with respect to the straight line $y=x$.

If one of the functions increases (decreases), then the other function increases (decreases).

Finding the Inverse Function

The equation $y=f(x)$ is solved with respect to the variable $x$.

From the obtained roots, those that belong to the interval $X$ are found.

The found $x$ are matched to the number $y$.

Example 1

Find the inverse function for the function $y=x^2$ on the interval $X=[-1,0]$

Since this function is decreasing and continuous on the interval $X$, then on the interval $Y=$, which is also decreasing and continuous on this interval (Theorem 1).

Let's calculate $x$:

\ \

Select suitable $x$:

Answer: inverse function $y=-\sqrt(x)$.

Problems on finding inverse functions

In this part we will consider inverse functions for some elementary functions. We will solve problems according to the scheme given above.

Example 2

Find the inverse function for the function $y=x+4$

Let's find $x$ from the equation $y=x+4$:

Example 3

Find the inverse function for the function $y=x^3$

Solution.

Since the function is increasing and continuous over the entire domain of definition, then, according to Theorem 1, it has an inverse continuous and increasing function on it.

Let's find $x$ from the equation $y=x^3$:

Finding suitable values ​​of $x$

The value is suitable in our case (since the domain of definition is all numbers)

Let's redefine the variables, we get that the inverse function has the form

Example 4

Find the inverse function for the function $y=cosx$ on the interval $$

Solution.

Consider the function $y=cosx$ on the set $X=\left$. It is continuous and decreasing on the set $X$ and maps the set $X=\left$ onto the set $Y=[-1,1]$, therefore, by the theorem on the existence of an inverse continuous monotone function, the function $y=cosx$ in the set $ Y$ there is an inverse function, which is also continuous and increasing in the set $Y=[-1,1]$ and maps the set $[-1,1]$ to the set $\left$.

Let's find $x$ from the equation $y=cosx$:

Finding suitable values ​​of $x$

Let's redefine the variables, we get that the inverse function has the form

Example 5

Find the inverse function for the function $y=tgx$ on the interval $\left(-\frac(\pi )(2),\frac(\pi )(2)\right)$.

Solution.

Consider the function $y=tgx$ on the set $X=\left(-\frac(\pi )(2),\frac(\pi )(2)\right)$. It is continuous and increasing on the set $X$ and maps the set $X=\left(-\frac(\pi )(2),\frac(\pi )(2)\right)$ onto the set $Y=R$, therefore, by the theorem on the existence of an inverse continuous monotone function, the function $y=tgx$ in the set $Y$ has an inverse function, which is also continuous and increasing in the set $Y=R$ and maps the set $R$ onto the set $\left(- \frac(\pi )(2),\frac(\pi )(2)\right)$

Let's find $x$ from the equation $y=tgx$:

Finding suitable values ​​of $x$

Let's redefine the variables, we get that the inverse function has the form

We have already encountered a problem where, given a given function f and a given value of its argument, it was necessary to calculate the value of the function at this point. But sometimes you have to face the inverse problem: to find, given a known function f and its certain value y, the value of the argument in which the function takes a given value y.

A function that takes each of its values ​​at a single point in its domain of definition is called an invertible function. For example, a linear function would be invertible function. But the quadratic function or the sine function will not be invertible functions. Since a function can take the same value with different arguments.

Inverse function

Let us assume that f is some arbitrary invertible function. Each number from the domain of its values ​​y0 corresponds to only one number from the domain of definition x0, such that f(x0) = y0.

If we now associate each value x0 with a value y0, we will obtain a new function. For example, for a linear function f(x) = k * x + b, the function g(x) = (x - b)/k will be its inverse.

If some function g at every point X range of values ​​of the invertible function f takes a value such that f(y) = x, then we say that the function g- there is an inverse function to f.

If we are given a graph of some invertible function f, then in order to construct a graph of the inverse function, we can use the following statement: the graph of the function f and its inverse function g will be symmetrical with respect to the straight line specified by the equation y = x.

If a function g is the inverse of a function f, then the function g will be an invertible function. And the function f will be the inverse of the function g. It is usually said that two functions f and g are mutually inverse to each other.

The following figure shows graphs of functions f and g mutually inverse to each other.

Let us derive the following theorem: if a function f increases (or decreases) on some interval A, then it is invertible. The inverse function g, defined in the range of values ​​of the function f, is also an increasing (or correspondingly decreasing) function. This theorem is called inverse function theorem.

Let us assume that we have a certain function y = f (x), which is strictly monotonic (decreasing or increasing) and continuous on the domain of definition x ∈ a; b ; its range of values ​​y ∈ c ; d, and on the interval c; d in this case we will have a function defined x = g (y) with a range of values ​​a ; b. The second function will also be continuous and strictly monotonic. With respect to y = f (x) it will be an inverse function. That is, we can talk about the inverse function x = g (y) when y = f (x) will either decrease or increase over a given interval.

These two functions, f and g, will be mutually inverse.

Why do we even need the concept of inverse functions?

We need this to solve the equations y = f (x), which are written precisely using these expressions.

Let's say we need to find a solution to the equation cos (x) = 1 3 . Its solutions will be all points: x = ± a rc c o s 1 3 + 2 π · k, k ∈ Z

For example, the inverse cosine and cosine functions will be inverse to each other.

Let's look at several problems to find functions that are inverse to given ones.

Example 1

Condition: what is the inverse function for y = 3 x + 2?

Solution

The domain of definitions and range of values ​​of the function specified in the condition is the set of all real numbers. Let's try to solve this equation through x, that is, by expressing x through y.

We get x = 1 3 y - 2 3 . This is the inverse function we need, but y will be the argument here, and x will be the function. Let's rearrange them to get a more familiar notation:

Answer: the function y = 1 3 x - 2 3 will be the inverse of y = 3 x + 2.

Both mutually inverse functions can be plotted as follows:

We see the symmetry of both graphs regarding y = x. This line is the bisector of the first and third quadrants. The result is a proof of one of the properties of mutually inverse functions, which we will discuss later.

Let's take an example in which we need to find the logarithmic function that is the inverse of a given exponential function.

Example 2

Condition: determine which function will be the inverse for y = 2 x.

Solution

For a given function, the domain of definition is all real numbers. The range of values ​​lies in the interval 0; + ∞ . Now we need to express x in terms of y, that is, solve the specified equation in terms of x. We get x = log 2 y. Let's rearrange the variables and get y = log 2 x.

As a result, we have obtained exponential and logarithmic functions, which will be mutually inverse to each other throughout the entire domain of definition.

Answer: y = log 2 x .

On the graph, both functions will look like this:



Basic properties of mutually inverse functions

In this paragraph we list the main properties of the functions y = f (x) and x = g (y), which are mutually inverse.

Definition 1

1. We already derived the first property earlier: y = f (g (y)) and x = g (f (x)).
2. The second property follows from the first: the domain of definition y = f (x) will coincide with the range of values ​​of the inverse function x = g (y), and vice versa.
3. The graphs of functions that are inverse will be symmetrical with respect to y = x.
4. If y = f (x) is increasing, then x = g (y) will increase, and if y = f (x) is decreasing, then x = g (y) will also decrease.

We advise you to pay close attention to the concepts of domain of definition and domain of meaning of functions and never confuse them. Let's assume that we have two mutually inverse functions y = f (x) = a x and x = g (y) = log a y. According to the first property, y = f (g (y)) = a log a y. This equality will be true only in the case of positive values ​​of y, and for negative values ​​the logarithm is not defined, so do not rush to write down that a log a y = y. Be sure to check and add that this is only true when y is positive.

But the equality x = f (g (x)) = log a a x = x will be true for any real values ​​of x.

Don't forget about this point, especially if you have to work with trigonometric and inverse trigonometric functions. So, a r c sin sin 7 π 3 ≠ 7 π 3, because the arcsine range is π 2; π 2 and 7 π 3 are not included in it. The correct entry will be

a r c sin sin 7 π 3 = a r c sin sin 2 π + π 3 = = a r c sin sin π 3 = π 3

But sin a r c sin 1 3 = 1 3 is a correct equality, i.e. sin (a r c sin x) = x for x ∈ - 1; 1 and a r c sin (sin x) = x for x ∈ - π 2 ; π 2. Always be careful with the range and scope of inverse functions!

• Basic mutually inverse functions: power functions

If we have a power function y = x a , then for x > 0 the power function x = y 1 a will also be its inverse. Let's replace the letters and get y = x a and x = y 1 a, respectively.

On the graph they will look like this (cases with positive and negative coefficient a):



• Basic mutually inverse functions: exponential and logarithmic

Let's take a, which will be a positive number not equal to 1.

Graphs for functions with a > 1 and a< 1 будут выглядеть так:



• Basic mutually inverse functions: trigonometric and inverse trigonometric

If we were to plot the main branch sine and arcsine, it would look like this (shown as the highlighted light area).

What is an inverse function? How to find the inverse of a given function?

Definition .

Let the function y=f(x) be defined on the set D, and E be the set of its values. Inverse function with respect to function y=f(x) is a function x=g(y), which is defined on the set E and assigns to each y∈E a value x∈D such that f(x)=y.

Thus, the domain of definition of the function y=f(x) is the domain of values ​​of its inverse function, and the domain of values ​​y=f(x) is the domain of definition of the inverse function.

To find the inverse function of a given function y=f(x), you need :

1) In the function formula, substitute x instead of y, and y instead of x:

2) From the resulting equality, express y through x:

Find the inverse function of the function y=2x-6.

The functions y=2x-6 and y=0.5x+3 are mutually inverse.

The graphs of the direct and inverse functions are symmetrical with respect to the straight line y=x(bisectors of the I and III coordinate quarters).

y=2x-6 and y=0.5x+3 - . The graph of a linear function is . To construct a straight line, take two points.



It is possible to express y unambiguously in terms of x in the case when the equation x=f(y) has a unique solution. This can be done if the function y=f(x) takes each of its values ​​at a single point in its domain of definition (such a function is called reversible).

Theorem (necessary and sufficient condition for the invertibility of a function)

If the function y=f(x) is defined and continuous on a numerical interval, then for the function to be invertible it is necessary and sufficient that f(x) be strictly monotonic.

Moreover, if y=f(x) increases on an interval, then the function inverse to it also increases on this interval; if y=f(x) decreases, then the inverse function decreases.

If the reversibility condition is not satisfied throughout the entire domain of definition, you can select an interval where the function only increases or only decreases, and on this interval find the function inverse to the given one.

A classic example is . In between)