Math Homework Help

Question

Find the domain of the function. Give your result in interval notation.

(a) f(x) = x-6/x^2-36

(b) f(x)== 4−√25−x^2

(c) f(x)=4/1-2ln x

Answer 1

Hi,

(a) f(x) = x-6/x^2-36

First, you need to know a²- b²= ( a-b) (a+b)

x^2-36 = ( x-6 ) ( x + 6 ) AND simplify the given fraction

f(x) = x-6/x^2-36 = ( x - 6 ) / ( x-6 ) ( x + 6 ) ------> f(x) = 1/ ( x + 6 )

Now domain is x + 6 ≠ 0 -----> x ≠-6 AND ALSO ( x-6 ) ≠ 0 ----> x ≠6

BE CAREFUL when you simplify any fraction must consider to the domain,

that expression that you simplified must be opposite ZERO, interval notation :

Domain (-∞,-6) , ( -6, 6) , (6, ∞)

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(b) f(x)== 4−√25−x^2 for finding domain, under the radical must be positive 25−x^2 ≥ 0 after solving this inequality Domain will be -5 ≤ x≤ 5 interval notation [-5 , 5 ]

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(c) f(x)=4/1-2ln x in fact this function is a fraction so denominator must be opposite ZERO and include logarithm that you know logarithm has defined only for positive number it means x≥ 0

Now: 1-2ln x ≠ 0 -----> ln x ≠ 1/2 -----> x ≠e^1/2 (1)

x ≥0 (2)

if you show these condition (1) and (2) on x-axis Domain will be [0 , e^1/2 ) , ( e^1/2 , ∞)

I hope it is useful,

Minoo