range of 1/(9+x^2)

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`x`²	`x`^□	log_□	√☐	^□√☐	≤	≥	□□	·	÷	`x`^◦	π
(☐)^′	`ddx`	∂∂`x`	∫	∫_□^□	lim	∑	∞	`θ`	(`f` ◦ `g`)	`f`(`x`)

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log_□

θ

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∫

ddx

≥	≤	·	÷	`x`^◦	(☐)	\|☐\|	(`f` ◦ `g`)	`f`(`x`)	ln	`e`^☐
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simplify solve for expand factor rationalize

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Steps Examples

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range 19+x²

Solution

0<f(x)≤19

Interval Notation

(0, 19 ]

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Solution steps

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Function range definition

The set of values of the dependent variable for which a function is defined

Rewrite as

19+x² =y

Multiply both sides by 9+x²

1=y(9+x²)

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

Discriminant 1=y(9+x²): −36y²+4y

−36y²+4y≥ 0 : 0≤ y≤ 19

Check if the range interval endpoints are included

y=0 : Excluded

y=19 : Included

Therefore the range is

0<f(x)≤ 19

Solve instead

domain 19+x²

inverse 19+x²

intercepts 19+x²

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