题目
题目

MATH 1151 AU2025 (18443) Calculus Concept Readiness Assessment (CCR)

单项选择题

Which of the following defines 𝑓 − 1 for 𝑓 ( 𝑡 ) = ln ⁡ ( 𝑡 + 2 ) ?

选项
A.𝑓 − 1 ( 𝑡 ) = 𝑒 𝑡 − 2
B.𝑓 − 1 ( 𝑡 ) = 𝑒 𝑡 + 2
C.𝑓 − 1 ( 𝑡 ) = 𝑒 𝑡 2
D.𝑓 − 1 ( 𝑡 ) = 𝑒 𝑡 + 2
E.𝑓 − 1 ( 𝑡 ) = 𝑒 𝑡 − 2
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标准答案
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思路分析
To start, we need to determine the inverse function of f(t) = ln(t + 2). This means solving the equation y = f(t) for t in terms of y, or more generally t = f^{-1}(y). Option 1: f^{-1}(t) = e^t − 2. This result comes from setting t inside the logarithm to x, so y = ln(x + 2). Exponentiating both sides gives e^y = x + 2, hence......Login to view full explanation

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类似问题

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

Question text Find the inverse function of the exponential function [math: f(x)=6e2x−4+1]\displaystyle f(x)={6\,e^{2\,x-4}+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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