# How To Solve Inverse Questions Quickly

## How to Solve Inverse Question Quickly:-

Inverse functions gives lots of troubles so here’s a quick run down of what an inverse function is, and how to find it.

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

## 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)

(e) None of these

#### Explanations

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

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

### 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 off(x). Then we solve for x and finally, swap x and y.

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

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

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

f-1 (x)= $\frac{5x+4}{3x-1} \$

## How To Solve Quickly Inverse Questions.

### Question 3

f(x) = log$_5$ (2x-1)

Options

(a) f-1 = $\frac{1}{2} \$ (5x+ 1)

(b) f-1 = $\frac{2}{5} \$ (2x+ 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.

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

Read Also: Formula for Inverse Question