How To Solve Inverse Questions Quickly

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)