Hey Chegg I need help with these 2 problems for precalculus

Hey Chegg, I need help with these 2 problems for precalculus.

1. Find the inverse function, f^-1 (explicit form) of the following: f (x) = 3^x+4 - 5.

2. A thermometer reading 99 degrees F is brought into a room with a constant temperature of 75 degrees F. If the thermometer 88 degrees F after 7 minutes, how long will it take for the thermometer to read 77 degrees F.

For problem 2, make sure to use Newton\'s Law of Cooling, which is u (t) = T + (u₀ - T)e^kt

For both problems, show your work. The more work that you show for both problems, it will help me to understand how to approach the problems. I need it for my assignment.

Solution

i) f (x) = 3^x+4 - 5

To find inverse plug x = y and y =x and solve for y

y = 3^(x+4) - 5

x = 3^(y+4) - 5

(x+5) = 3^(y+4)

Take log on both sides:

ln(x+5) = (y+4)ln(3)

ln(x+5)/ln3 = y +4

y = ln3(x+5) - 4

f^-1(x) = ln3(x+5) - 4

2) u(t) = T + (uo -T)e^kt

U, temperature after time t
T, temperature of the environment
Uo, temperature initially

88 = 75 + (99 -75)e^7k

13 = 24e^7k

ln(13/24) = e^7k ;k = -0.0875

Aftre hwo much time u(t) = 77

77 = 75 + (99 - 75)e^kt

2/24= e^kt

ln(2/24) = kt

t = ln(2/24)/-0.0875

t = 28.39 minutes