How do you find the inverse of #f(x)=2x+3# and is it a function?

Question

How do you find the inverse of  #f(x)=2x+3# and is it a function?

Madelyn Anderson · Accepted Answer

# f^-1 x = 1/2(x - 3 )#

Basically this is an exercise in changing the subject of an expression. Given y in terms of x , the inverse is to express x in terms of y.

let y = 2x + 3 - now change the subject to x.

hence : 2x = y - 3 → # x = 1/2(y - 3 ) #

It is usual to write the inverse in terms of x.

# rArr f^-1 x = 1/2(x - 3)#

This is the equation of a line and is a function

Liam Brennan · Answer

undefined To find the inverse of $ f(x) = 2x + 3 $, follow these steps:

1. Replace $ f(x) $ with $ y $.
2. Swap the variables $ x $ and $ y $.
3. Solve the resulting equation for $ y $.
4. Replace $ y $ with $ f^{-1}(x) $ to express the inverse function.

Step 1: $ f(x) = 2x + 3 $ becomes $ y = 2x + 3 $.

Step 2: Swap the variables: $ x = 2y + 3 $.

Step 3: Solve for $ y $: 
$$
x = 2y + 3 \
2y = x - 3 \
y = \frac{x - 3}{2}
$$

Step 4: Replace $ y $ with $ f^{-1}(x) $:
$$
f^{-1}(x) = \frac{x - 3}{2}
$$

To determine if $ f^{-1}(x) $ is a function, we need to check if it passes the vertical line test. If any vertical line intersects the graph of $ f^{-1}(x) $ at more than one point, then it is not a function.

Since $ f^{-1}(x) = \frac{x - 3}{2} $ is a linear function, it passes the vertical line test. Therefore, $ f^{-1}(x) $ is indeed a function.