Part 3: The function f(x) = 2* is reflected vertically, translated 5 units up and 3 units left to find g(x). (10 points per answer)
1. Write out the function g(x).
2. What is the domain of g(x)?
3. What is the range of g(x)?

First, let's define the transformations in a general way.

Vertical translation:

A vertical translation of N units is defined as:

g(x) = f(x) + N

if N is positive, the translation is up

if N is negative, the translation is down.

Horizontal translation:

A horizontal translation of N units can be written as:

g(x) = f(x - N)

if N is positive, the translation is to the right

if N is negative, the translation is to the left.

Reflected vertically (or a reflection over the x-axis)

This type of reflection changes the sign of the y component.

We can write those as:

g(x) = -f(x)

Then:

We start with the function:

f(x) = 2^x

First we reflect it vertically.

g(x) = -2^x

now we do a vertical translation of 5 units up, then:

g(x) = -2^x + 5

Now we do a translation of 3 units to the left, then we get:

g(x) = - 2^(x + 3) + 5

Then we already did the first step.

Now we want to find the domain for this function, the domain will be the set of the possible values of x that we can input in the function g(x).

Notice that in this function the denominator is never zero, so there are no values of x that generate problems here, so the domain will be the set of all real numbers, that can be written as:

(-∞, ∞)

3) Now we want to find the range, this is the set of the possible values of g(x).

The Largest value of g(x) will be when the exponential part tends to zero, this is for really small values of x, such that when x goes to minus infinity, -2^(x + 3) goes to zero.

Then:

g(-∞) ≈ 5

The maximum of the range is 5.

And we do not have a minimum, because as x increases the function will decrease infinitely.

Then the range will be:

(-∞, 5)

(A graph of the function can be seen below)

answer: 1 and 3

step-by-step explanation:

step-by-step explanation:

part a. the differences are decreasing, so we know the sequence is not arithmetic. the ratios are constant (=3/4), so we know the sequence is geometric.

part b. from part a, we know the common ratio is 3/4, so each term is 3/4 of the one before. the initial term is the number of gallons at h=1.

g[h] = (3/4)·g[h-1]

g[1] = 384 . . . . initial value required for a recursive definition

part c. the general formula for a geometric sequence is

a[n] = a[1]·r^(n-1)

we have a[1] = 384, r = 3/4, and the term number given by h. the formula that follows is

g[h] = 384·(3/4)^(h-1)

part d. filling in the value and doing the arithmetic, we have

g[12] = 384·(3/4)^11 ≈ 384·0.042235136 ≈ 16.2183 . . gallons

part e. each gallon is 4 quarts, so

16.2183 gal × 4 qt/gal = 64.8732 qt