Find the Domain and Range f(x)=(8x)/(x^2-64)

Problem

ƒ(x)=(8*x)/(x2−64)

Solution

1. Identify the domain by finding values of x that make the denominator zero.

x2−64=0

2. Solve for x by factoring the difference of squares.

(x−8)*(x+8)=0

x=8,x=−8

3. State the domain as all real numbers except those that cause division by zero.

Domain: *{[x∈ℝ,x≠8,x≠−8]}

4. Identify the range by analyzing the horizontal asymptote and the behavior of the function.

(lim_x→∞)((8*x)/(x2−64))=0

5. Determine if the function reaches the asymptote by setting the function equal to the horizontal asymptote value y=0

(8*x)/(x2−64)=0

8*x=0

x=0

6. Conclude the range based on the fact that the function is continuous on its intervals and passes through y=0 covering all real values.

Range: *{y∈ℝ}

Final Answer

Domain: *(−∞,−8)∪(−8,8)∪(8,∞), Range: *(−∞,∞)

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