# How To Solve Inverse Questions Quickly

## 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 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.

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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