Question

NO LINKS!!

Find the function value, if possible. (If the answer is undefined, enter UNDEFINED.)
f(x) = |x|/x

a. f(x -1)
{ _____ if x < _____
{ _____ if x > _____

Answer 1

`f(x-1) \begin{cases}-1\;\; \text{if}\;\;x < 1\\\;\:\:1\;\; \text{if}\;\;x > 1\end{cases}`

Explanation:

Given function:

`f(x)=(|x|)/(x)`

Therefore:

`f(x-1)=(|x-1|)/(x-1)`

The function is undefined when the denominator is zero.

Therefore, the function is undefined when x = 1.

As the numerator is an absolute value it is always positive.

`\textsf{When}\;\;x < 1 \implies f(-x-1)=(|-x-1|)/(-x-1)=(|-(x+1)|)/(-(x+1))=(|x+1|)/(-(x+1))=-1`

`\textsf{When}\;\;x > 1 \implies f(x-1)=(|x-1|)/(x-1)=1`

Answer 2

• 
  `f(x - 1) = \left \{ {{-1 \ if \ x < 1} \atop {1 \ if \ x > 1}} \right.`

-----------------------------------

Given function:

• f(x) = |x|/x

The function f(x - 1) is:

• f(x - 1) = |x - 1|/(x - 1)

The absolute value is always positive in this case and can't be zero otherwise it is undefined.

When x - 1 > 0, both the numerator and denominator are positive with same value so:

• f(x - 1) = 1 if x - 1 > 0 or x > 1

When x - 1 < 0, the numerator is positive and denominator is negative with same value so:

• f(x - 1) = - 1 if x - 1 < 0 or x < 1