a7- Describe and interpret domains and ranges using set notation 

Gabrielle

Explanation:

The domain of a function or graph states all possible x values for that function.

The range of a function sates all possible y values for that function.

Set notation is a way of writing range and domain.

   set notation is enclosed in curly brackets like this one.

It is useful to know what the different parts of an exponential function represent when you're trying to determine its range and domain.

  ( b is to the power of (x/c) )

a= starting value

b= common ratio

c= time it takes for starting value to change by the common ratio

d= horizontal asymptote

ex.

This is a graph of an exponential function with a horizontal asymptote of 0 or no horizontal asymptote. Its y values will never go below 0.

In this graph the y values will continue to grow infinitely. You can tell it is exponential growth because its common ratio is larger than 1.

There are no reflections in the x or y axis.

The range of this function can be represented using set notation as follows:

This is read: y is an element of the real numbers, such that y is greater than 0.

The domain of the function can be represented using set notation as follows:

This states that x is an element of the real numbers. It places no other limitations on the x values, showing that any real number could be an x value on this graph. The domain of exponential functions is always the same, as seen above.

I will now demonstrate how to write the domain or range of a function from an equation rather than a graph.

ex.

We can see that the horizontal asymptote of this equation is -4, and that there are no reflections through the x or y axis. We can also see that the common ratio is greater than zero making this a growth curve. From these pieces of information we can determine that the y values of this equation will be greater than negative 4. This is written in set notation as follows : 

Often times the easiest way to determine the range of an exponential function is to put the equation in general form and put it into your graphing calculator. This is because when there is a negative applied to the x value of the equation it causes a reflection through the y axis, and when there is a negative applied to the starting value of the equation it causes a reflection through the x axis of the equation. This can sometimes be confusing, and difficult to picture in your head. Graphing the function often helps!

Since a logarithm is the inverse of an exponent the domain and ranges are also switched.

Where in an exponential function the domain is always  in a logarithmic equation it is the range which is always .

An exponential function's inverse(logarithmic) function has the same domain and range, only switched. (the domain values are that of the range and the range values become those of the domain).

The inverse function's graph is a reflection of the original graph across the line y=x.

Problems:

1.

graph taken from www.analyzemath.com

determine the range and domain of the equation represented by the graph above.

2. Determine the range of the following equation:

Solutions:

1. We can see that this graph demonstrates exponential growth, and has a horizontal asymptote of 0.

    We can then determine that the range of the function is

    The domain of the function is, as always

2. We can see that this is a decay curve with a reflection through the x axis. It also has a horizontal asymptote of 7.

     Because it's a decay curve reflected, it's going to be growing, but only up until 7, the horizontal asymptote. This means that y is increasing until 7, making y any real number less than 7.

    The range of the equation is: