Question

Find the vertical and horizontal asympotes, domain, range, and roots of f(x)=. - 1/x-3 +2

please show all steps because I’m so lost

Answer 1

The function has a vertical asymptote at x = 3, a horizontal asymptote at y = 2, a domain of all real numbers except x = 3, a range of all real numbers except y = 2, and a root at x = 3.5.

Step-by-step explanation:

To find the vertical and horizontal asymptotes, domain, range, and roots of the function f(x) = -1/(x-3) + 2, we follow several steps.

• Vertical asymptote: This occurs where the function is undefined, which is when the denominator equals zero. For our function, this happens at x = 3, so our vertical asymptote is x = 3.
• Horizontal asymptote: As x approaches infinity, the term -1/(x-3) approaches zero, so the function approaches y = 2. Thus, the horizontal asymptote is y = 2.
• Domain: The function is defined for all real numbers except where the denominator is zero. Therefore, the domain is x ≠ 3, or all real numbers except 3.
• Range: Given the horizontal asymptote at y = 2, and that the function can take values arbitrarily large or small except for 2, the range is all real numbers y ≠ 2.
• Roots: To find the roots, we set the function equal to zero and solve for x. After setting f(x) = 0, rearrange to get 1/(x-3) = 2. Cross-multiplication yields x - 3 = 1/2, so the root is x = 3.5 or x = 7/2.