Inverse Functions

An inverse function or an anti function is defined as a function which can reverse another function. In simple words, if any function “f” takes x to y then, the inverse of “f” i.e. “f-1” will take y to x. Learn more about the inverse of functions in detail from here. The concepts that are covered include:

  • What are Inverse Functions?
  • How to Find the Inverse of a Function?
  • Common Inverse Functions
    • Inverse of Trigonometric Functions
    • Inverse of Rational Functions
    • Inverse of Hyperbolic Functions
    • Inverse Logarithmic Functions and Inverse Exponential Functions
  • Example Questions from Inverse of Functions
  • Inverse of Functions Worksheet

What is an Inverse Function?

A function accepts values, performs particular operations on these values and generates an output. The inverse function accepts the resultant, performs an operation and reaches back to the original function.

If you consider f and g are inverse functions, f(g(x)) = g(f(x)) = x. A function that consists of its inverse functions fetches the original value.

Important Inverse Function Notes:

The inverse of a function – The relation that is developed when the independent variable is interchanged with the variable that is dependent in a specified equation and this inverse may or may not be a function.

Inverse function – If the inverse of a function is a function by itself, then it is known as inverse function, denoted by f-1(x).

How to Find the Inverse of a Function?

Generally, the method of calculating an inverse is swapping of coordinates x and y. This newly created inverse is a relation but not necessarily a function.

The original function has to be a one-to-one function to assure that its inverse will be also a function. A function is said to be a one to one function only if every second element corresponds to the first value (values of x and y are used only once).

Inverse Functions

Inverse Functions

You can apply on the horizontal line test to verify whether a function is a one-to-one function. If a horizontal line intersects the original function in a single region, the function is a one-to-one function and inverse is also a function.

Common Inverse Functions

The inverses of some of the most common inverse functions are given below. Even the inverse function of trigonometric function, rational function, hyperbolic function and log functions are also discussed in the following points. First, the list of common inverse function is given which include:

Function

Inverse Function

Comment

+

*

/

Don’t divide by 0

1/x

1/y

x and y not equal to 0

x2

√y

x and y >= 0

xn

y1/n

n is not equal to 0

ex

ln(y)

Y > 0

ax

loga(y)

y and a > 0

Sin (x)

Sin-1 (y)

– π/2 to + π/2

Cos (x)

Cos-1 (y)

0 to π

Tan (x)

Tan-1 (y)

– π/2 to + π/2

Inverse of Trigonometric Functions

The inverse trigonometric functions are also known as arc function as they produce the length of the arc which is required to obtain that particular value. There are six inverse trigonometric functions which include arcsine (sin-1), arccosine (cos-1), arctangent (tan-1), arcsecant (sec-1), arccosecant (cosec-1), and arccotangent (cot-1). To learn more about the inverse trigonometric functions along with their graphs, follow the linked article.

Inverse Rational Function

A rational number is a number of form f(x) = P(x)/Q(x) where Q(x) is ≠ 0. To find the inverse of a rational number, follow the following steps. An example is also given below which can help to understand the concept better.

  • Step 1: Replace f(x) = y
  • Step 2: Interchange x and y
  • Step 3: Solve for y in terms of x
  • Step 4: Replace y with f-1(x) and the inverse function is obtained.

Example:

Find the inverse for the function f(x) = \(\frac{3x+2}{x-1}\)

Solution:

First, replace f(x) with y and the function becomes,

y = \(\frac{3x+2}{x-1}\)

By replacing x with y we get,

x = \(\frac{3y+2}{y-1}\)

Now, solve y in terms of x :

x (y – 1) = 3y + 2

=> xy – x = 3y +2

=> xy – 3y = 2 + x

=> y (x – 3) = 2 + x

=> y = (2 + x) / (x – 3)

So, y = f-1(x) = \(\frac{x+2}{x-3}\)

Inverse Hyperbolic Functions

Just like the inverse trigonometric function, the inverse hyperbolic functions are the inverses of the hyperbolic functions. There are 6 main inverse hyperbolic functions which include sinh-1, cosh-1, tanh-1, csch-1, coth-1, and sech-1. Check out inverse hyperbolic functions formula to learn more about these function in detail.

Inverse Logarithmic Functions and Inverse Exponential Function

The natural log functions are inverse of the exponential functions. Check the following example to understand the inverse exponential function and logarithmic function in detail. Also, get more insights of how to solve similar questions and thus, develop problem-solving abilities.

Example:

Find the inverse of the function f(x) = Ln(x – 2)

Solution:

First, replace f(x) with y

So, y = Ln(x – 2)

Replace the equation in exponential way , x – 2 = ey

Now, solving for x,

x = 2 + ey

Now, replace x with y and thus, f-1(x) = y = 2 + ey

Example Question from Inverse Function

To solve for an equation:

f(x) = 2x + 3, at x = 4

We have,

f(4) = 2 x 4 + 3

f(4) = 11

Now, let’s apply for reverse on 11.

f-1(11) = (11 – 3) / 2

f-1(11) = 4

Magically we get 4 again

Therefore, f-1(f(4)) = f(4)

So, when we apply function f and its reverse f-1 gives the original value back again, i.e, f-1(f(x)) = x

An example illustrating inverse functions using Algebra.

Put “y” for “f(x)” and solve for x:

The function:

f(x)

=

2x+3

Put “y” for “f(x)”:

y

=

2x+3

Subtract 3 from both sides:

y-3

=

2x

Divide both sides by 2:

(y-3)/2

=

x

Swap sides:

x

=

(y-3)/2

Solution (put “f-1(y)” for “x”) :

f-1(y)

=

(y-3)/2

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Practise This Question

Pavan drew two lines and he measured some angles. He found that 1=5 and so he concluded that the two lines are parallel.

State whether his conclusion is true or false.