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blagie [28]
2 years ago
11

Find the domain of the function (f/g)(x) given f (x)=x^2-4x-5 and g(x)=x^2-25

Mathematics
1 answer:
stealth61 [152]2 years ago
5 0

Answer:

all real numbers except ±5.

Step-by-step explanation:

Each function (f(x), g(x)) has a domain that is all real numbers. Their quotient (f(x)/g(x)) must exclude values that make g(x) = 0. The quotient is undefined when the denominator is zero. Those excluded values are x = ±5.

The domain of (f/g)(x) is all real numbers except ±5.

___

Of course you recognize x^2 -25 = 0 has solutions x = ±√25 = ±5. You can get there two ways:

  1. add 25 and take the square root: x^2=25; x=±√25.
  2. factor the difference of squares and set the factors to zero: (x-5)(x+5)=0 has solutions x-5=0 and x+5=0, that is, x = ±5.
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\rule{130}1

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\rule{130}1

☯ \underline{\boldsymbol{According\: to \:the\: Question\:now :}}

:\implies\sf x^2 + (x + 2)^{2} = 514 \\\\\\:\implies\sf x^2 + x^2 + 4x + 4 = 514\\\\\\:\implies\sf 2x^2 + 4x + 4 = 514 \\\\\\:\implies\sf 2x^2 + 4x = 514 - 4\\\\\\:\implies\sf 2x^2 + 4x = 510 \\\\\\:\implies\sf 2x^2 + 4x - 510 = 0\\\\\\:\implies\sf x^2 + 2x - 255 = 0 \:\:\:\:\:\Bigg\lgroup \bf{Dividing\:by\:2}\Bigg\rgroup\\\\\\:\implies\sf x^2 + 17x - 15x - 255 = 0\\\\\\:\implies\sf x(x + 17) - 15(x + 17) = 0\\\\\\:\implies\sf (x + 17) (x - 15)\\\\\\:\implies\sf x = - 17\:or\:x = 15\\\\\\:\implies\underline{\boxed {\sf x = 15}}\:\:\:\:\:\Bigg\lgroup \bf{Positive\:odd\: number}\Bigg\rgroup

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\bullet\: \textsf {x = } \textbf {15}

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