How To Solve Inverse Questions Quickly

March 6, 2024

How to Solve Inverse Question Quickly in Aptitude

Here , In this Page you will find, How to Solve Inverse Questions Quickly. Inverse functions has Many importance in Aptitude. It is also used in Geometry.

For a function to have an inverse, the function has to be 1 to 1.  That means every output only has one input. In Algebraic Questions Algebra is involved whereas In Geometric Questions Trigonometry is involved.

How to Solve Inverse Questions Quickly

How to solve Inverse function

Inverse Rational Function

A rational function is a function of form f(x) = P(x)/Q(x) where Q(x) ≠ 0. To find the inverse of a rational function, follow the following steps. 

  • 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 of the function is obtained.

Inverse Questions and Answers

Question 1 : Find an equation for the inverse of the function given f(x) = 3x + 2

Options

A. f-1(x) = \frac{1}{3} \ (x-2)

B.  f-1(x) = \frac{1}{2} \ (x-2)

C. f-1(x) = \frac{3}{2} \ (3x-2)

D,  f-1(x) = \frac{1}{2} \ (2x-1)

Explanations :

First we drop the function notation and write y instead of f(x). Then we solve for x and finally, swap x and y.

y = 3x+2

swap  x and y

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

Correct Options (A)

Question 2 : Find the inverse of f(x) = \frac{x+4}{3x – 5}

Options

A.  f-1(x) = \frac{5x+4}{3x-1} \

B.  f-1(x) = \frac{5x+3}{3x-1} \

C.  f-1(x) = \frac{2x+3}{5x-1} \

D. None of these

Explanations :

First we drop the function notation and write y instead of f(x). Then swap x and y.

y = \frac{x+4}{3x-5}

x = \frac{y+4}{3y-5}   (multiply by 3y-5)
x(3y-5) = y+4   (distribute)
3xy – 5x = y+4   (add 5x, subtract y)
3xy – y = 5x +4  (factor out y)
y(3x – 1) = 5x + 4  (divide by 3x – 1).

y = \frac{5x+4}{3x-1} \

f-1 (x)= \frac{5x+4}{3x-1} \

Correct option (A)

Question 3 : Find the inverse of f(x) = log _5 (2x-1)

Options

A . f^{-1} = \frac{1}{2} (5^x + 1)

B. f^{-1} = \frac{2}{5} (2^x + 1)

C. f^{-1} =  \frac{3}{5} (x+ 1)

D. None of these

Explanations :

y = log _5 (2x-1)   (re-write it as exponential statement.

5^{y} = 2x – 1

Swap x and y

5^{x} = 2y – 1(add 1)
5^{x} + 1 = 2y (divide by 2)
\frac{5^x + 1}{2} \ = y
y = \frac{1}{2}(5^x + 1)
f-1 = \frac{1}{2} (5^x + 1)

Correct Options (A)

Question 4 : What  is the inverse function of tan(x)?

Options

A. cos^{-1}(x)

B. tan^{-1}(x)

C. tan^{-1}(\frac{1}{x})

D. None

Solution :

y = tan(x)

x = tan(y)

y = tan^{-1}(x)

Correct Option B

Question 5:  What is the value of sin(sin^{-1}(x)) where x belongs to [-1,1]

Options

A. 1

B. 0

C. x

D. None

Solution:

In this Expression sin(sin^{-1}(x))

Domain of sin^{-1}(x) is [-1,1] so it’s value will be x

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