6.2 Exponential functions (in class)

Let [math]f(x)=a\cdot b^x[/math] be an exponential function where [math]a[/math] and [math]b[/math] are positive numbers and [math]b>1[/math]. If [math]g(x)=-2f(-x)+1[/math],

(a) What is the [math]y[/math]-intercept of [math]g(x)[/math]?
(A) [math]1-2a[/math]
(B) [math]1[/math]
(C) [math]-2a[/math]
(D) [math]2-2a[/math]

(b) What is the asymptote of [math]g(x)[/math]?
(A) [math]y=0[/math]
(B) [math]y=1[/math]
(C) [math]y=-1[/math]
(D) [math]y=3[/math]
(E) [math]y=-3[/math]

Let [math]f(x)=a\cdot b^x+c[/math] be an exponential function where [math]a, b[/math] and [math]c[/math] are real numbers and [math]0<b<1[/math]. If the range of [math]f(x)[/math] is [math](-\infty, c)[/math], whcih of the following statements is/are true?

(Calculator) The smell of canned herring is infamous. Japanese media NHK used instruments to measure the odor of canned herring. The odor value was as high as 8000Au (about 20 times that of Taiwanese stinky tofu). Assuming that in a well-ventilated room, the half-life of the odor of canned herring is 8 hours. At 12:00, Ms. Fly opened a can in her room. How long will it take until, in hours (integer), the odor in her room is the same as Taiwanese stinky tofu?

The function [math]f(t)=100\cdot0.95^t[/math] represents the relationship between the time [math]t[/math] (in years) and the number of capybaras in a forest. If the unit of [math]t[/math] is changed from years to months, what will be the new function?

(Calculator) Eliza invented a new kind of drug. She wants to find the half-life of the drug. Mr. Fang tried the new drug. Suppose the concentration of the drug in Mr. Fang’s body was 15ppm at 9AM. One hour later, the concentration became 12ppm, what is the half-life, in hours, of the drug?

(Calculator) Savings insurance is a kind of investment. It is similar to a savings account. You deposit money into an “insurance account” to earn interest. The interest rates of insurance accounts are always higher than normal savings accounts, but there are still drawbacks. That is, the insurance account will restrict you from withdrawing your money for a certain number of years (usually 10 to 20 years, depending on your contract). If you terminated before the contracted number of years, you will have to pay a penalty.

(a) Mr. Fang signed a 10-year savings insurance contract with a principal of NTD500,000 in the insurance account with a 10-year contract. The interest rate is from 2.2% to 2.5%. Suppose Mr. Fang decides to withdraw 15 years later. How much will Mr. Fang get at minimum? Round to the nearest integer.

“Inflation” is a general progressive increase in the prices of goods and services. For example, a certain product costs 10 dollars this year, but you will have to spend 12 dollars to buy the same product the next year. At the same time, if your salary increases, then the effect of inflation will be eliminated. Otherwise, you will feel that the prices of goods and services are becoming more and more unaffordable.
(b) Mr. Fang wants to purchase a car that costs NTD900,000. Suppose the price of the car raises 1% per year because of inflation. Given that Mr. Fang plans to purchase the car 15 years later, how much money should he prepare in addition to his savings insurance? Round to the nearest integer.

Find the number of the solutions of the equation [math]-2x+1=2^{-x}[/math]

What is the minimum integer value of [math]f(x)=4^x+3\cdot2^{x+1}+5[/math]?