题目
题目

MCD2130 - T2 - 2025 10.7 Online Quiz - MCD2130_QUIZ_NINE Circular Functions, Continuity and Differentiability

单项选择题

MTH1010_09_04_2

选项
A.a. a)
B.b. b)
C.c. c)
D.d. d)
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思路分析
We are asked to select the function which has an inverse function. Option a: f : R → R where f(x) = sin(x). Over the entire real line, the sine function is not one-to-one because it repeats values in every 2π interval. Therefore it is not invertible as a function from R to R. Option b: f : (0, 2π) → R where f(x) = sin(x). Although this interval i......Login to view full explanation

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Find the domain and range of the inverse of the given function. f(x) =

Problem: Find the inverse for the function 𝑓 ( 𝑥 ) = 3 𝑥 + 1 2 𝑥 − 4 .  Suppose a student writes the following solution. Is it correct? Line 1:   Write 𝑦 = 3 𝑥 + 1 2 𝑥 − 4 and solve for 𝑥 .  Line 2:    𝑦 ( 2 𝑥 − 4 ) = 3 𝑥 + 1   Line 3:   2 𝑥 𝑦 − 4 𝑦 = 3 𝑥 + 1   Line 4:   2 𝑥 𝑦 − 3 𝑥 = 1 + 4 𝑦   Line 5:   𝑥 ( 2 𝑦 − 3 ) = 1 + 4 𝑦   Line 6:   𝑥 = 1 + 4 𝑦 2 𝑦 − 3   Line 7:   Now interchange variables:  𝑦 = 1 + 4 𝑥 2 𝑥 − 3     Line 8:   Therefore   𝑓 − 1 ( 𝑥 ) = 1 + 4 𝑥 2 𝑥 − 3      Line 9:   Simplify:   𝑓 − 1 ( 𝑥 ) = 𝑥 + 4 2 − 3 𝑥       a) Is the above solution correct? If not, in which line is the first error? (If there is more than one error, select the line where the FIRST error occurs.) [ Select ] First error is in Line 2. First error is in Line 9. The solution is correct. First error is in Line 1. First error is in Line 7. First error is in Line 4. First error is in Line 6. First error is in Line 3. First error is in Line 5. First error is in Line 8. b) What is the correct final answer to the problem? [ Select ] y = (x+1) / (2-3x) y = (1+4x) / (1-6x) y = (x+4) / (2-3x) None of the other options y = (1+4x) / (2x-3) c) What is the domain of 𝑓 − 1 ( 𝑥 ) ? [ Select ] All real numbers except 3/2 All real numbers except -4 All real numbers except -1/4 All real numbers except 1/6 All real numbers except 2/3

Question text Find the inverse function of the exponential function [math: f(x)=2e5x−2+1]\displaystyle f(x)={2\,e^{5\,x-2}+1}. Note: Type [math: ln(a)] to enter [math: loge(a)]log_{e}(a). [math: f−1(x)=]f^{-1}(x)=[input] Check Question 7

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