For which interval is f(x) = \dfrac{1}{x - 2} negative?

Question

BlackTom AI · Accepted Answer

The function given is f(x) = 1/(x - 2). First, note its domain excludes x = 2 because the denominator would be zero there.

- Option a: (2, ∞). In this interval, x - 2 > 0, so f(x) = 1/(positive) is positive, not negative. This makes this interval incorrect.

- Option b: (-∞, 1). This interval is entirely to the left of 1, which is still left of 2, so in this region x - 2 < 0 and f(x) would be negative. However, this option does not include all x < 2; it stops at 1 and omits the portion (1, 2). Therefore, it does not represent all x where f(x) is negative, so it is not correct.

- Option c: (-∞, 2). Here every x less than 2 yields x - 2 < 0, so f(x) = 1/(x - 2) is negative throughout this entire interval. This matches the set of all x for which f(x) is negative, making this option correct.

- Option d: (-∞, ∞). This includes x = 2 (where the function is undefined) and also includes values greater than 2 where the function is positive. Since the interval includes points where the function is not negative, this choice is incorrect.

In summary, because the sign of f(x) is determined by the sign of x - 2 (with the numerator fixed at 1), the function is negative exactly for all x < 2, i.e., on the interval (-∞, 2). The other options either miss portions of this interval or include regions where f(x) is not negative.

MHF4U - Advanced Functions 12 (2025-26) - A

For which interval is f(x) = \dfrac{1}{x - 2} negative?

查看解析

登录即可查看完整答案

类似问题

Find the [math: y] intercept of [math: f(x)=4x2+36x+3+−7] f(x)= \frac{4x^2+3}{6x+3} +-7 Give your answer correct to 2 decimal places.

Computing Rational Functions For the given functions and , do the following computation. Choose the best answer. and Find

Computing Rational Functions For the given functions f(x) and g(x), do the following computation. Choose the best answer. f(x)= 5x+1 8x−1 and g(x)= 3x 8x−1 Find f g (x)

Computing Rational Functions For the given functions 𝑓 ( 𝑥 ) and 𝑔 ( 𝑥 ) , do the following computation. Choose the best answer. 𝑓 ( 𝑥 ) = 5 𝑥 + 1 8 𝑥 − 1 and 𝑔 ( 𝑥 ) = 3 𝑥 8 𝑥 − 1 Find 𝑓 𝑔 ( 𝑥 )

The graph below shows a rational function.Which of the following best describes the graph’s equation, domain, and horizontal asymptote?

Given the following features of a rational function:x-intercept: -2 y-intercept: -3 vertical asymptote: x = 2horizontal asymptote: y = 3 Which of the following is the correct equation?

The range of 𝑦 = 9 7 − 𝑥

Choose which graph best represents the curve [math: y=4−1x−2]y=4 - \dfrac{1}{x-2} .

更多留学生实用工具

考试浏览器助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单