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How do you find the domain and range of #f(x)=(x+7)/(x^2-49)#?

Answer 1

Genesis Allen

The domain is #x in RR-{7}#
The range is #f(x) in (-oo,0^-)uu(0^+,+oo)#

Let's make the function simpler.

#f(x)=(x+7)/(x^2-49)=cancel(x+7)/(cancel(x+7)(x-7))#

#f(x)=1/(x-7)#

The denominator is #!=0#

Consequently,

#x-7!=0#, #=>#,#x!=7#

The domain is #I=RR-{7}#

The function is bijective over the domain #I#

#f(-oo)=1/(-oo-7)=0^-#

#f(7^-)=1/(7^(-) -7)=-oo#

#f(7^+)=1/(7^(+) -7)=+oo#

#f(+oo)=1/(+oo-7)=0^+#

Consequently,

The range is #f(x) in (-oo,0^-)uu(0^+,+oo)#

graph{(1/(x-7)) [-10.06, 10.22, 27.2, -13.34]}

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Answer 2

Avery Alexander

To find the domain, identify values of x that make the denominator nonzero. The domain is all real numbers except x ≠ ±7.

To find the range, analyze the behavior of the function as x approaches positive and negative infinity. The range is all real numbers except y ≠ 0.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.