What is the domain of the following function? $$f(x) = \sqrt {\frac {19 x}{x^{2} - 36}}$$

Question

BlackTom AI · Accepted Answer

The problem asks for the domain of f(x) = sqrt(19x / (x^2 - 36)).
First, remember that a square root requires its radicand to be nonnegative and the denominator of the fraction cannot be zero.
Identify the critical points where the expression could change sign or be undefined: x = -6, x = 0, and x = 6.
We will test the sign of the fraction 19x / (x^2 - 36) across the intervals determined by these points: (-∞, -6), (-6, 0), (0, 6), (6, ∞).
Interval (-∞, -6): choose x = -7. Numerator 19x is negative; denominator x^2 - 36 = 49 - 36 = 13 is positive. Negative over positive is negative, so not allowed.
Interval (-6, 0): choose x = -1. Numerator 19x is negative; denominator (-1)^2 - 36 = 1 - 36 = -35 is negative. Negative over negative is positive, so this interval is allowed. At x = -6 the denominator is zero (not allowed), so -6 is excluded; at x = 0 the numerator is zero, giving zero inside the square root, which is allowed, so 0 is included. Thus this interval contributes (-6, 0].
Interval (0, 6): choose x = 1. Numerator positive; denominator 1^2 - 36 = -35 is negative. Positive over negative is negative, so not allowed.
Interval (6, ∞): choose x = 7. Numerator positive; denominator 49 - 36 = 13 positive. Positive over positive is positive, so this interval is allowed; 6 itself is excluded due to division by zero at x = 6.
Combining the allowed intervals and including the endpoint 0, the domain is (-6, 0] ∪ (6, ∞).
Note: The prompt’s answer options aren’t provided here, but the derived domain matches the standard domain given by ensuring the radicand is nonnegative and the denominator is nonzero.

MATH1061/1002/1021 (ND) MATH1061 Canvas Quiz 10

What is the domain of the following function? \[f(x) = \sqrt {\frac {19 x}{x^{2} - 36}}\]

查看解析

登录即可查看完整答案

类似问题

Part 1Find the domain of the function f(x)equals=StartFraction x squared minus 4 Over x squared minus 9 EndFractionx2−4x2−9. Part 1The domain is [input]enter your response here .(Type your answer in interval notation.)

Find the Domain Find the domain of the function. g(x)=x+3x2−81\displaystyle g\left(x\right)=\frac{x+3}{x^2-81} The domain is (−∞,(-\infty,[Fill in the blank] )∪()\cup([Fill in the blank] ,[Fill in the blank] )∪()\cup([Fill in the blank] ,∞),\infty)

Find the domain of the function. Write your answer in interval notation. Use inf for .

State the domain of the following function in the graph below. (K:1)

What is the domain for the function;

In a consumer society, many adults channel creativity into buying things

Economic stress and unpredictable times have resulted in a booming industry for self-help products

People born without creativity never can develop it

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