What is the range of f(x)=5csc(1/3 x)-1 ?

The range of f(x)=5csc(1/3 x)-1 is f(x) = 4

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range of f(x)=5csc(1/3 x)-1

range

f (x) = 5 csc (\frac{1}{3} x) - 1

f (x) \leq - 6 or f (x) \geq 4

Interval Notation

(- \infty, - 6] \cup [4, \infty)

Solution steps

The range of the basic

csc

function is

csc (\frac{1}{3} x) \leq - 1 or csc (\frac{1}{3} x) \geq 1

csc (\frac{1}{3} x) \leq - 1 or csc (\frac{1}{3} x) \geq 1

Multiply the edges of the range by:

5

5 csc (\frac{1}{3} x) \leq - 5 or 5 csc (\frac{1}{3} x) \geq 5

Add to the edges of the range:

- 1

5 csc (\frac{1}{3} x) - 1 \leq - 6 or 5 csc (\frac{1}{3} x) - 1 \geq 4

Therefore the range is

f (x) \leq - 6 or f (x) \geq 4

(1, 2), (5, 6)

\frac{2}{x - 2} - 8

\frac{3}{x + 2} - x - 3

ln (x - 1) - 1

f (x) = \frac{4 x ^{2} + 4 x - 1}{2 x ^{3} + 8 x ^{2}}

What is the range of f(x)=5csc(1/3 x)-1 ?
The range of f(x)=5csc(1/3 x)-1 is f(x)<=-6\lor f(x)>= 4