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

Question

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

BlackTom AI · Accepted Answer

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 x = e^y − 2. Replacing y with t yields f^{-1}(t) = e^t − 2. This matches the correct algebraic steps.

Option 2: f^{-1}(t) = e^t + 2. This would imply x + 2 = e^y becomes x = e^y − 2 still, but adding 2 instead of subtracting 2 would not satisfy the original relation. The sign change breaks the inverse relationship, so this is incorrect.

Option 3: f^{-1}(t) = e^t 2. This is not a proper expression; it appears to misplace formatting or omit an operator between e^t and 2. As written, it is not a valid function form and does not represent the inverse.

Option 4: f^{-1}(t) = e^t + 2. This is the same issue as Option 2 but with the plus sign. It would yield x = e^y + 2, which does not invert the original ln(t + 2) correctly, so it’s incorrect.

Option 5: f^{-1}(t) = e^t − 2. This is the same as Option 1 in value, but since the question presents multiple choices, having a duplicate correct-looking option could be confusing. It still represents the correct inverse form, but in many contexts only one option should be correct.

In summary, the correct inverse function is f^{-1}(t) = e^t − 2, derived by solving y = ln(x + 2) for x and substituting back. The other options either alter the sign, introduce formatting issues, or fail to satisfy the inversion relationship, leading to incorrect inverses.

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

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

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