The main idea is that an inverse function exists only when each output comes from a unique input. That one-to-one (injective) property guarantees that you can unambiguously recover the original input from the output. If a function assigns the same value to two different inputs, there isn’t a single preimage for that output, so you can’t define a single inverse value as a function. For example, x^2 maps both 1 and -1 to 1, so it cannot have a proper inverse on all of R. On the other hand, a function like x^3 assigns each real number to a unique real number, so it has an inverse. Continuity or differentiability don’t ensure invertibility, because they don’t prevent multiple inputs from producing the same output. Surjectivity would guarantee that every real number is hit, but even then you’d still need injectivity to have a well-defined inverse function.

The area of a rectangle is found by multiplying its length by its width. With area 72 and length 8, set up 72 = 8 × w. Solve for w by dividing both sides by 8: w = 72 ÷ 8 = 9. This works because 8 × 9 equals 72. Checking other values shows they don’t produce 72 (for example, 8 gives 64, 6 gives 48, 12 gives 96), so the width must be 9.

The device that provides electrical power for an aircraft while it’s on the ground is called a Ground Power Unit, abbreviated as GPU. It supplies the aircraft with the necessary electrical power—often 400 Hz AC or DC power—so systems like lights, avionics, air conditioning, and preflight equipment can run without starting the engines or the APU. GPUs are used during preflight checks, maintenance, and boarding to keep everything powered safely on the ramp. The other terms shown aren’t devices that power the aircraft on the ground: FAA is the Federal Aviation Administration, the regulatory body; FLT is shorthand for flight; FAK isn’t a standard aviation power device. So GPU is the correct match for the ground power source.

In the 24-hour clock, hours run from 00 to 23, with a leading zero for single-digit hours. Times before noon keep the same hour value, just written as two digits. So 7:00 AM becomes 07:00, written without a colon as 0700. The other options correspond to 5:00 AM (0500), 9:00 AM (0900), and noon (1200). Therefore, the 24-hour representation is 0700.