Domain and transformations of exponential and logarithmic functions

Algebra

All exponential functions have as domain the set of real numbers because the domain is the set of numbers that can enter the function and enable to produce a number as output. In exponential functions whatever real number can be operated. (-infinity, infinity)

You have ln (x+4) so everything is shifted by 4. The domain of ln (x+4) is now -4 < x < infinity (Shifting infinity by a finite number gives you infinity again.) So,-4 < x < infinity is the domain of ln (x+4).

(2 [less than] t [less than] infinity)

For your function f (t) = 5.5exp (t) the function is continuous for all values of t as exp (t) is continuous for all values t, i.e. The domain of the function is -oo < t < oo

2a. subtracting 3 on the inside the function moves it 3 units to the right, that's the only transformation.

the vertical asymptote results from an invalid input to the function, like dividing by zero or taking the square root of a negative number, or in this case, taking the log of zero. so we need to find the value where x-3=0, thus the asymptote occurs at x=3. The x intercept is when y (or g (x) in this case)=0-0=g (x)=log (x-3) log (x-3)=0 x-3=10^0 x-3=1 x=4 so the intercept is at (4,0)

2b. We see that the right side of g (x), which is -log (x), is the result of multiplying the right side of f (x) by -1, so by rule 5, the graph of g (x) is the graph of f (x) reflected across the x-axis.

Flipped over the x-axis;

horizontal asymptote y=0

no x-intercept

3a. The principal amount (P) is 3000, since that is what was originally deposited.

The rate (r) is 0.06 because 6% means 6/100.

The number (n) that it is compounded is 1, since annually means only once a year.

The time (t) is 9 years; Therefore when plugged into the equation, the return is $5,068.44

3b. We can just plug in the numbers. P, r, and t are the same, but now n changes from 1 to 4. As a result, the return comes out to be $5,127.42

3c. Compounding quarterly yields more interest. This is because when we do it once a year, it only multiplies the whole thing once by 1.06. When we do it four times a year, it multiplies it by 1.015^4, which is 1.06136355, which is actually more than 1.06.