What is the domain of the following function?
𝑓
(
𝑥
)
=
3
𝑥
𝑥
2
−
25

Question

BlackTom AI · Accepted Answer

For f(x) to be defined as a real number, the expression inside the square root must be ≥ 0 and any denominators must be nonzero. Here the radicand is 3x/(x^2 − 25), so we require
3x/(x^2 − 25) ≥ 0, with x ≠ ±5 (denominator zero).

1. Relevant concepts
- A square root requires its argument ≥ 0.
- Rational expressions are undefined where the denominator is 0.
- Solve a rational inequality by factoring and using a sign chart.

2. Reasoning and algebra
Write the inequality and factor the denominator:
3x/(x^2 − 25) ≥ 0
x^2 − 25 = (x − 5)(x + 5)
So 3x/(x − 5)(x + 5) ≥ 0.

Critical points where numerator or denominator is zero: x = −5, 0, 5. Denominator zeros (x = ±5) are excluded from the domain. Use intervals determined by these points: (−∞, −5), (−5, 0), (0, 5), (5, ∞).

3. Sign chart (evaluate sign of 3x and each factor)
- Interval (−∞, −5): x < −5 → 3x negative; (x+5) negative; (x−5) negative → denominator = (neg)(neg)=positive → fraction negative → not allowed.
- Interval (−5, 0): x between −5 and 0 → 3x negative; (x+5) positive; (x−5) negative → denominator negative → fraction (neg)/(neg)=positive → allowed.
- At x = 0: numerator 0 → fraction = 0 → allowed (√0 = 0).
- Interval (0, 5): x between 0 and 5 → 3x positive; (x+5) positive; (x−5) negative → denominator negative → fraction negative → not allowed.
- Interval (5, ∞): x > 5 → 3x positive; (x+5) positive; (x−5) positive → denominator positive → fraction positive → allowed.

4. Why the correct answer is correct
Allowed x-values are (−5, 0] and (5, ∞). Endpoints x = ±5 are excluded because the denominator is zero; x = 0 is included because the radicand equals 0 (permissible).

5. Why other options are incorrect
- Options using ±11 are irrelevant (they come from misfactoring x^2 − 25) and therefore wrong.
- The option with (−5,0)∪[5,∞) is wrong because it excludes x = 0 (which yields radicand 0 and is allowed) and incorrectly includes x = 5 (denominator zero, excluded).

6. Conclusion
The domain is (−5, 0] ∪ (5, ∞), which matches the provided correct answer.

What is the domain of the following function? 𝑓 ( 𝑥 ) = 3 𝑥 𝑥 2 − 25单项选择题

类似问题

What is the domain of the following function? f(x)= √ 18x x2−64

Find the maximal (implied) domain of the function with rule [math: f(x)=3x−2+3].

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 .

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

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

What is the domain for the function;

更多留学生实用工具

智能学习助手

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