Find the Domain and Range f(x)=(x^2+x-2)/(x^2-3x-4)

Problem

ƒ(x)=(x2+x−2)/(x2−3*x−4)

Solution

1. Factor the numerator and the denominator to identify potential discontinuities and simplify the expression.

ƒ(x)=((x+2)*(x−1))/((x−4)*(x+1))

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

(x−4)*(x+1)=0

x=4,x=−1

3. State the domain as all real numbers except for the vertical asymptotes.

Domain:(−∞,−1)∪(−1,4)∪(4,∞)

4. Determine the horizontal asymptote by comparing the degrees of the numerator and denominator. Since both are degree 2, the asymptote is the ratio of the leading coefficients.

y=1/1=1

5. Check if the function ever crosses the horizontal asymptote by setting ƒ(x)=1

(x2+x−2)/(x2−3*x−4)=1

x2+x−2=x2−3*x−4

4*x=−2

x=−0.5

Since the function crosses y=1 at x=−0.5 the value y=1 is included in the range.

6. Analyze the range by observing the behavior around the vertical asymptotes. Because the factors in the denominator are not squared and do not cancel with the numerator, the function will approach both positive and negative infinity, covering all real values.

Range:(−∞,∞)

Final Answer

Domain:{x|x≠−1,4},Range:{y|y∈ℝ})

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