Sketch the curve y = 1 / x – 3

Question

Sketch the curve y = 1 / x &#8211; 3

Suresh Chand · Accepted Answer

Solution Given that (y = frac{1}{x - 3}) A. Domain Clearly, domain is the set of all real number except 3. i.e. domain is (-∞, 3) ∪ (3, ∞) B. Intercept For x-intercept: Put y = 0 we get x = ∞ For y-intercept: Put x = 0  we get (frac{-1}{3}) Thus curve meet the x-axis and y-axis at origin C. Symmetry Since f(-x) ≠ -f(x), so it is not symmetrical about origin. D. Asymptotes (lim_{x to pm ∞} f(x) = lim_{x to pm ∞} frac{1}{x-3} = ∞) Thus the curve doesn't have horizontal assymptotes (lim_{x to 3} frac{1}{x-3} = ∞) So, x = 3 is the vertical assymptotes. E. Extreme (f^{'}(x) = frac{d}{dx} (x-3)^{-1}) (f^{'}(x) = -1 (x-3)^{-1-1}) (f^{'}(x) = frac{1}{(x-3)^{2}}) (f^{'}(x) = -frac{1}{(x-3)^2}) For critical points (f^{'}(x) = 0) and (f^{'}(x) = ∞) We get x = 3 Interval (-∞, 3) (3,∞) Sign of f'(x) -ve +ve Nature of f(x) Decreasing Increasing F. Concavity For concavity and point of inflection (f^{''}(x) = 2 times frac{1}{(x-3)^3}) For point of inflection (f^{'}(x) = 0) and (f^{'}(x) = ∞) We get x = 3 Interval (-∞, 3) (3,∞) Sign of (f^{''}(x)) -ve -ve Nature of f(x) Concave Down Concave Down G. Summary Interval (-∞, 3) (3,∞) Sign of f(x) Decreasing Increasing f(x) Concave Down Concave Down Using this table with information we can sketch the graph as follows:

Sketch the curve \(y = \frac{1}{x – 3}\)

Leave your Answer:

Hits Counter

Question

Sketch the curve \(y = \frac{1}{x – 3}\)

Leave your Answer: