Question

I need to find what f1 and f2 equals

Answer 1

Absolute value is defined as

$|x| = \begin{cases}x & \text{if }x\ge0 \\ -x&\text{if }x<0\end{cases}$

By this definition, we have

$|2x-3| = \begin{cases} 2x-3 &\text{if }2x-3 \ge0 \\ -(2x-3) &\text{if }2x-3<0\end{cases}$

or, after some simplifying,

$|2x-3| = \begin{cases} 2x-3 &\text{if }x \ge\frac32 \\ 3-2x &\text{if }x<\frac32\end{cases}$

So a = 3/2.

If x < 3/2, we get the second case:

$f_1 = 1 + |2x-3| \\\\ f_1 = 1 + (3-2x) \\\\ \boxed{f_1 = 4 - 2x}$

If x ≥ 3/2, we get the first case:

$f_2 = 1 + |2x-3| \\\\ f_2 = 1 + (2x-3) \\\\ \boxed{f_2 = 2x-2}$