What is the range of f(x)=log_{10}(5x-x^2-6) ?

The range of f(x)=log_{10}(5x-x^2-6) is f(x)<=-2log_{10}(2)

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range of f(x)=log_{10}(5x-x^2-6)

range

f (x) = lo g_{10} (5 x - x^{2} - 6)

f (x) \leq - 2 lo g_{10} (2)

Interval Notation

(- \infty, - 2 lo g_{10} (2)]

Solution steps

The function range is the combined domain of the inverse functions

Find the inverse function(s) of:

lo g_{10} (5 x - x^{2} - 6)

Inverse of

lo g_{10} (5 x - x^{2} - 6) : - \frac{- 5 + 1 - 5 ^{x} \cdot 2 ^{x + 2}}{2}, - \frac{- 5 - 1 - 5 ^{x} \cdot 2 ^{x + 2}}{2}

- \frac{- 5 + 1 - 5 ^{x} \cdot 2 ^{x + 2}}{2}, - \frac{- 5 - 1 - 5 ^{x} \cdot 2 ^{x + 2}}{2}

Find the domain of each inverse function

Domain of

- \frac{- 5 + 1 - 5 ^{x} \cdot 2 ^{x + 2}}{2} : x \leq - 2 lo g_{10} (2)

Domain of

- \frac{- 5 - 1 - 5 ^{x} \cdot 2 ^{x + 2}}{2} : x \leq - 2 lo g_{10} (2)

Combine the intervals

f (x) \leq - 2 lo g_{10} (2)

y = 6 x + 2

x^{4} - x^{2}

y = \frac{1}{4} cot (π x)

y = 3 x - 2, (2, 5)

f (x) = 2 - 8 e^{x}

What is the range of f(x)=log_{10}(5x-x^2-6) ?
The range of f(x)=log_{10}(5x-x^2-6) is f(x)<=-2log_{10}(2)