Find the Domain and Range f^-1

Problem

ƒ(x)=(x+2)/(x−3)

Solution

1. Identify the relationship between a function and its inverse. The domain of ƒ(x)(−1) is the range of ƒ(x) and the range of ƒ(x)(−1) is the domain of ƒ(x)

2. Find the domain of ƒ(x) by identifying values that make the denominator zero.

x−3=0

x=3

The domain of ƒ(x) is all real numbers except x=3 which is the range of ƒ(x)(−1)

3. Find the range of ƒ(x) by finding the horizontal asymptote. For a rational function where the degrees of the numerator and denominator are equal, the asymptote is the ratio of the leading coefficients.

y=1/1

y=1

The range of ƒ(x) is all real numbers except y=1 which is the domain of ƒ(x)(−1)

4. Verify by finding the inverse function ƒ(x)(−1) Swap x and y and solve for y

x=(y+2)/(y−3)

x*(y−3)=y+2

x*y−3*x=y+2

x*y−y=3*x+2

y*(x−1)=3*x+2

ƒ(x)(−1)=(3*x+2)/(x−1)

5. Determine the domain of ƒ(x)(−1) from its formula. The denominator is zero when x=1

x−1=0

x=1

6. Determine the range of ƒ(x)(−1) from its formula. The horizontal asymptote is the ratio of leading coefficients.

y=3/1

y=3

Final Answer

Domain of *ƒ(x)(−1):{[x∈ℝ,x≠1]}, Range of *ƒ(x)(−1):{[y∈ℝ,y≠3]}

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