give the range and the domain in

**interval notation**of f(x)= 1/x^2-8-
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give the range and the domain in **interval notation** of f(x)= 1/x^2-8

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Is the denominator of the function (x^{2}-8), or is it x^{2}?

As written, with parenthesis for clarification, the function is:

f(x) = (1/x^{2}) - 8

The domain of the function is the set of real number values that we may input into the function and get a real number value as an output. Since division by 0 is undefined, the domain of the function (as written) is all real numbers except 0.

If we compare this function to (1/x^{2}), which has a range of (0, infinity). Now the function that you wrote has been vertically shifted down 8 units, and the range would be (-8,infinity)

If, instead, you meant:

f(x) = 1/(x^{2}-8), we will again consider where the denominator may equal 0, and exclude these values from the domain.

x^{2} - 8 = 0

x^{2} = 8 (taking the square root of both side and evaluating the result)

x = ±2√(2)

Then the function has two vertical asymptotes at these values (x=2√(2) and x= - 2√(2)), and the domain is all real numbers except these two values.

Since the highest power of x in the numerator is less than the highest power of x in the denominator, there is a horizontal asymptote at y=0.

The range of this function is all real numbers not equal to 0.

The only special points are those when denominator is zero. Then the function is not defined and those points shall be excluded from domain.

x2-8=0;

x=±2√2;

Thus domain, D, of a function is D:x∈(-∞;-2√2)∪(-2√2;2√2)∪(2√2;∞).

For the range:

Consider the denominator. Since x2 is always non-negative, x2-8 is always greater or equal to -8. When the x approaches -2√2 from the right of 2√2 from the left, denominator is negative and goes to zero, therefore the whole function goes to minus infinity. So on the interval (-2√2;2√2) the function f(x)=1/(x2-8) goes from -1/8 to -∞. On two other intervals the function goes to +infinity when x approaches -2√2 from the left or 2√2 from the right. When x goes to ±∞, denominator goes to +∞ and the whole function, f(x), goes to zero, but never attains that value. Thus, the range of this function is:

R: f(x)∈(-∞;-1/8]∪(0;∞).

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