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[fiacre68]

Consider the following functions: y = 2^x, y= 3^x and y=0.5^x

1. For each function, describe key properties relating to domain, range, intercepts, increasing/decreasing intervals, and asymptotes.

2. Make comparisons between the graphs, and explain the relationship between the y-intercepts.

Thank you guys! Cheers~~!

[eoforwic]

Dear Dong Wook,

y=2^(x)

The domain of the rational expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.
All real numbers
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y=2^(x)

Since x is on the right-hand side of the equation, switch the sides so it is on the left-hand side of the equation.
2^(x)=y

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.
ln(2^(x))=ln(y)

The left-hand side of the equation is equal to the exponent of the logarithm argument because the base of the logarithm equals the base of the argument.
xln(2)=ln(y)

Divide each term in the equation by ln(2).
(xln(2))/(ln(2))=(ln(y))/(ln(2))

Cancel the common factor of ln(2) in (xln(2))/(ln(2)).
x=(ln(y))/(ln(2))

The domain of an expression is all real numbers except for the regions where the expression is undefined. This can occur where the denominator equals 0, a square root is less than 0, or a logarithm is less than or equal to 0. All of these are undefined and therefore are not part of the domain.
y<=0

The domain of the rational expression is all real numbers except where the expression is undefined.
y$0_(-<Z>I<z>,0) U (0,<Z>I<z>)

The domain of the inverse of y=2^(x) is equal to the range of f(y)=(ln(y))/(ln(2)).
Range: y$0_(-<Z>I<z>,0) U (0,<Z>I<z>)
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Intercect is 1 for all, that answer question 2
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y=2^(x)

The domain of the rational expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.
All real numbers

A horizontal asymptote can potentially be found by finding the limit of the function as the value approaches <Z>I<z>.
L[x:<Z>I<z>,2^(x)]

The limit of 2^(x) as x approaches <Z>I<z> is <Z>I<z>
L[x:<Z>I<z>,2^(x)]=<Z>I<z>

The value of L[x:<Z>I<z>,2^(x)] is <Z>I<z>.
<Z>I<z>

There are no horizontal asymptotes because the limit does not exist.
No horizontal asymptote approaching <Z>I<z>.

A horizontal asymptote can potentially be found by finding the limit of the function as the value approaches -<Z>I<z>.
L[x:-<Z>I<z>,2^(x)]

The limit of 2^(x) as x approaches -<Z>I<z> is 0
L[x:-<Z>I<z>,2^(x)]=0

The value of L[x:-<Z>I<z>,2^(x)] is 0.
0

The horizontal asymptote is the value of y as x approaches -<Z>I<z>.
y=0

Since there is no remainder from the polynomial division, there are no oblique asymptotes.
No Oblique Aymptotes

This is the set of all asymptotes for y=2^(x).
No Vertical Asymptotes_Horizontal Aysmptote:y=0_No Oblique Aysmptotes
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[roche]

Domain: all real numbers
range: y>0

Here is an on-line graphing calculatorto help you with the rest
http://www.meta-calculator.com/online/