This question tests understanding of inverse functions within exponential and logarithmic contexts, a key concept in AP Precalculus. Inverse functions reverse the operations of the original function, such as logarithms being the inverse of exponentials. In this problem, we need to find the inverse of f(x) = $3·2^x$ - 5 by solving for x in terms of y. Starting with y = $3·2^x$ - 5, we add 5 to both sides to get y + 5 = $3·2^x$, then divide by 3 to get (y + 5)/3 = $2^x$, and finally apply log base 2 to get x = log₂((y + 5)/3). Choice B is correct because when we swap x and y, we get f^(-1)(x) = log₂((x + 5)/3), which properly reverses all operations. Choice A is incorrect because it subtracts 5 instead of adding it, failing to properly reverse the original function's operations. To help students: Emphasize the step-by-step process of finding inverses algebraically, and practice verifying inverses by checking that f(f^(-1)(x)) = x.

This question tests understanding of inverse functions within exponential and logarithmic contexts, a key concept in AP Precalculus. The range of an inverse function equals the domain of the original function. For f(x) = 2^(x+5), the domain is all real numbers (-∞,∞), which means the range of f^(-1) is also all real numbers. Choice B is correct because logarithmic functions (the inverse of exponentials) have range (-∞,∞) when their domain is (0,∞). Choice A is incorrect because it confuses domain and range - (0,∞) would be the domain of f^(-1), not its range. To help students: Create tables showing how domain and range swap for inverse function pairs, and reinforce that exponential functions have domain (-∞,∞).

This question tests understanding of inverse functions within exponential and logarithmic contexts, a key concept in AP Precalculus. Inverse functions reverse the operations of the original function, such as natural logarithms being the inverse of exponential functions with base e. To find the inverse of f(x) = e^(x-4), we solve y = e^(x-4) for x: ln(y) = x - 4, so x = ln(y) + 4. Choice C is correct because f^(-1)(x) = ln(x) + 4 properly reverses the original function's operations. Choice A is incorrect because it subtracts 4 instead of adding it, failing to account for the proper inverse relationship. To help students: Work through the algebraic steps systematically, and verify inverses by composing functions to check that f(f^(-1)(x)) = x.

This question tests understanding of inverse functions within exponential and logarithmic contexts, a key concept in AP Precalculus. To find the inverse of an exponential function with a vertical shift, we must carefully reverse all operations in order. For f(x) = $e^x$ - 7, we solve y = $e^x$ - 7 for x: adding 7 gives y + 7 = $e^x$, then taking natural log yields x = ln(y + 7). Choice B is correct because f^(-1)(x) = ln(x + 7) properly adds 7 before taking the logarithm, reversing the original subtraction. Choice A is incorrect because ln(x) - 7 would be the inverse of e^(x+7), not $e^x$ - 7, showing confusion about the order of operations. To help students: Use function composition to verify inverses, checking that f(f^(-1)(x)) = x, and emphasize how transformations affect inverse functions differently.

This question tests understanding of inverse functions within exponential and logarithmic contexts, a key concept in AP Precalculus. In real-world applications like compound interest, finding inverse relationships means solving for different variables. From A = $P(1+r)^t$, to solve for t: A/P = $(1+r)^t$, then ln(A/P) = t·ln(1+r), so t = ln(A/P)/ln(1+r). Choice B is correct because it properly uses the change of base formula and isolates t correctly. Choice C is incorrect because it multiplies the logarithms instead of dividing, which doesn't correctly isolate t. To help students: Connect abstract inverse concepts to practical applications, and practice solving exponential equations using logarithms.

This question tests understanding of inverse functions within exponential and logarithmic contexts, a key concept in AP Precalculus. Inverse functions reverse the operations of the original function, such as logarithms being the inverse of exponentials. In this problem, we need to find the inverse of f(x) = $3·2^x$, which involves isolating x from y = $3·2^x$. Choice B is correct because solving y = $3·2^x$ for x gives: y/3 = $2^x$, then x = log₂(y/3), so f^(-1)(x) = log₂(x/3). Choice C is incorrect because it represents another exponential function rather than a logarithmic inverse. To help students: Emphasize that exponential functions have logarithmic inverses, practice the algebraic steps of finding inverses, and verify by checking that f(f^(-1)(x)) = x.

This question tests understanding of inverse functions within exponential and logarithmic contexts, a key concept in AP Precalculus. When a function is restricted, its inverse's domain corresponds to the restricted function's range. For f(x) = $2^x$ restricted to x ≥ 1, we need to find the range: when x = 1, f(1) = 2¹ = 2, and as x increases, f(x) increases without bound. Choice C is correct because the range of the restricted function is [2,∞), which becomes the domain of f^(-1). Choice A is incorrect because it represents the domain of the unrestricted inverse function. To help students: Graph restricted functions to visualize their ranges, and emphasize how restrictions affect both domain and range of inverse functions.