# What is the linear function f if f(1-x)=3x+2?

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### 2 Answers

It is given that f(1-x)=3x+2 and that f(x) is a linear function.

Express 3x + 2 as an expression in terms of 1 - x

3x + 2

= 3x - 3 + 5

= 5 - 3 + 3x

= 5 - 3(1 - x)

As f(1 - x) = 3x+2 = 5 - 3(1 - x)

f(x) = 5 - 3x

The linear function f(x) = 5 - 3x

We'll recall the standard form of a linear function:

f(x) = ax + b

Since we know the determined form of f(1-x), we'll replace x by (1-x) in the standard form of linear function.

f(1-x) = a(1-x) + b

We'll remove the brackets:

f(1-x) = a - ax + b

But f(1-x) = 3x + 2

Comparing, we'll get:

a - ax + b = 3x + 2

-a = 3 => a = -3

a + b = 2 => -3 + b = 2 => b = 2 + 3 => b = 5

**The required linear function is: f(x) = -3x + 5.**