SOLUTION: 1. For the functions f(x) = 2^x and g(x) = 3^x : a) What are the Domains and Ranges of these functions? b) What point(s) do they have in common and why? c) Starting with y

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Question 1194670: 1. For the functions f(x) = 2^x and g(x) = 3^x :
a) What are the Domains and Ranges of these functions?
b) What point(s) do they have in common and why?
c) Starting with y = 2^x, what would y = 2^x+4 - 3 look like in comparison to y = 2^x? Explain each step and show each interim equation which illustrates each step. You may attach a graph if you'd like.
d) How does y = (1/2)^x look in relation to y = 2^x? Why is this?
Answer by greenestamps(13367) (Show Source):
You can put this solution on YOUR website!

Both functions are monotonically increasing -- an increase in x means an increase in y.

a) A positive number raised to any real number power is valid expression; and it is always positive. So the domain of both functions is all real numbers, and the range of both functions is y>0.

b) Since both functions are monotonically increasing, they will intersect at only one point. Since any number to the 0 power is equal to 1, the single point of intersection is (0,1).

c) I will assume by "2^x+4-3" what you mean is "2^(x+4)-3" = . The "x+4" shifts the graph 4 units to the left; then the "-3" shifts it 3 units down.

Here is a graph:
y=2^x (red),
y=2^(x+4) (green) (red, shifted 4 to the left),
y=2^(x+4)-3 (blue) (green, shifted 3 down)

d)

That means the graph of y=(1/2)x is the reflection in the y axis of y=2^x.

Here is a graph:
y=2^x (red);
y=(1/2)^x=2^(-x) (green)