What is the range of f(x)=((|x-5|))/((x-3)) ?

The range of f(x)=((|x-5|))/((x-3)) is f(x) = 0

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range of f(x)=((|x-5|))/((x-3))

range

f (x) = \frac{( ∣ x - 5 ∣ )}{( x - 3 )}

f (x) < - 1 or f (x) \geq 0

Interval Notation

(- \infty, - 1) \cup [0, \infty)

Solution steps

Find the minimum and maximum value in each defined interval and unite the results

Domain of

\frac{∣ x - 5 ∣}{x - 3} : x < 3 or x > 3

Extreme Points of

\frac{∣ x - 5 ∣}{x - 3} :

Minimum

(5, 0)

Absolute value functions are piecewise functions

Find the function intervals:

x < 3, 3 < x < 5, x \geq 5

Evaluate the function behavior for each interval

Find the range for the interval

- \infty < x < 3 : - \infty < f (x) < - 1

Find the range for the interval

3 < x < 5 : 0 < f (x) < \infty

Find the range for the interval

5 \leq x < \infty : 0 \leq f (x) < 1

Combine the ranges of all domain intervals to obtain the function range

f (x) < - 1 or f (x) > 0 or 0 \leq f (x) < 1

f (x) < - 1 or f (x) \geq 0

f (x) = 5 - x

y = 4, x = 5

f (x) = - 9 x - 2

f (x) = 5 x - 9

f (x) = \frac{3 x - 7}{x + 1}

What is the range of f(x)=((|x-5|))/((x-3)) ?
The range of f(x)=((|x-5|))/((x-3)) is f(x)<-1\lor f(x)>= 0