What is the range of f(x)=-x^4+4x^2-4 ?

The range of f(x)=-x^4+4x^2-4 is f(x)<= 0

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range of f(x)=-x^4+4x^2-4

range

f (x) = - x^{4} + 4 x^{2} - 4

f (x) \leq 0

Interval Notation

(- \infty, 0]

Solution steps

The function range is the combined domain of the inverse functions

Find the inverse function(s) of:

- x^{4} + 4 x^{2} - 4

Inverse of

- x^{4} + 4 x^{2} - 4 : 2 - - x, - 2 - - x, - x + 2, - - x + 2

2 - - x, - 2 - - x, - x + 2, - - x + 2

Find the domain of each inverse function

Domain of

2 - - x : - 4 \leq x \leq 0

Domain of

- 2 - - x : - 4 \leq x \leq 0

Domain of

- x + 2 : x \leq 0

Domain of

- - x + 2 : x \leq 0

Combine the intervals

f (x) \leq 0

f (x) = - x^{2} + 2 x + 5

f (x) = \frac{( x - 2 ) ^{2}}{x - 1}

f (x) = \frac{6 x + 1}{9 x ^{2} + 1}

f (x) = \frac{x ^{2} + 1}{x ^{2} + 5 x + 6}

f (x) = 3 x^{2} + 5 x - 2

What is the range of f(x)=-x^4+4x^2-4 ?
The range of f(x)=-x^4+4x^2-4 is f(x)<= 0