What is the range of f(x)=((x-3)^2)/(x^2) ?

The range of f(x)=((x-3)^2)/(x^2) is f(x)>= 0

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range of f(x)=((x-3)^2)/(x^2)

range

f (x) = \frac{( x - 3 ) ^{2}}{x ^{2}}

f (x) \geq 0

Interval Notation

[0, \infty)

Solution steps

Rewrite as

\frac{( x - 3 ) ^{2}}{x ^{2}} = y

Multiply both sides by

x^{2}

(x - 3)^{2} = y x^{2}

The range is the set of y for which the discriminant is greater or equal to zero

Discriminant

(x - 3)^{2} = y x^{2} : 36 y

36 y \geq 0 : y \geq 0

Check if the range interval endpoints are included

y = 0 :

Included

Therefore the range is

f (x) \geq 0

f (x) = ln (1 + \frac{( x + 1 )}{x + 4})

f (x) = x^{5} + 5 x^{4}

f (x) = 6 x + 9

f (x) = x^{3} - 7

f (x) = \frac{6 x}{x ^{2} + 9}

What is the range of f(x)=((x-3)^2)/(x^2) ?
The range of f(x)=((x-3)^2)/(x^2) is f(x)>= 0