How To Solve Inverse Questions Quickly

How to Solve Inverse Question Quickly in Aptitude

Here , In this Page How to Solve Inverse Questions Quickly is given.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.

Inverse questions are basically of 2 types.

  • Algebraic Questions
  • Geometric Questions

In Algebraic Questions Algebra is involved whereas In Geometric Questions Trigonometry is involved.

How to Solve Inverse Questions Quickly

Types 1. Algebraic Questions

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 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 3y – 1).

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

f-1 (x)= \frac{5x+4}{3x-1} \

Correct option (A)

Question 3 f(x) = log _5 (2x-1)

Options

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)

Types 2. Geometric Questions

Question 1 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 2  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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