document.write( "Question 1203719: Create an equation of a piecewise function that has the following properties:\r
\n" ); document.write( "\n" ); document.write( "- includes 4 different types of functions (e.g. linear, quadratic, exponential, square root)
\n" ); document.write( "- is continuous
\n" ); document.write( "- has limiting behaviour as x --> ± ∞, y --> + ∞ \n" ); document.write( "
Algebra.Com's Answer #839540 by greenestamps(13200)\"\" \"About 
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\n" ); document.write( "Assuming the four types of equations are in fact linear, quadratic, exponential, and square root....

\n" ); document.write( "(1) exponential and square root functions are monotonically increasing where they are defined. Since the behavior must be y goes to infinity as x goes to negative infinity, the leftmost part of the graph must be either linear (with a negative slope) or quadratic.
\n" ); document.write( "(2) a linear equation is probably the easiest to fit with the other types of functions, so let's not start with a linear function. So we will make the leftmost part of the graph quadratic. \"y=x%5E2\" is a simple one to work with.
\n" ); document.write( "(3) (1,1) is a solution to both \"y=x%5E2\" and \"y=sqrt%28x%29\", so it looks as if making the second function equal to \"y=sqrt%28x%29\" will be convenient. We can stop using that function at a convenient point, say (4,2).
\n" ); document.write( "(4) Making the third function linear will make it easy to make the \"junction\" between it and the fourth (exponential) function continuous. So choose a simple linear function containing (4,2): \"y=x-2\". We can end it again at a convenient point, say (5,3).
\n" ); document.write( "(5) Then the fourth function is exponential containing (5,3). Although it doesn't look \"nice\", the function \"y=e%5Ex%2B%283-e%5E5%29\" will do the job.

\n" ); document.write( "One of an infinite number of possibilities....

\n" ); document.write( "\"y=x%5E2\" for x less than 1 [red in the graph below]
\n" ); document.write( "\"y=sqrt%28x%29\" for x greater than or equal to 1 and less than 4 [green]
\n" ); document.write( "\"y=x-2\" for x greater than or equal to 4 and less than 5 [blue]
\n" ); document.write( "\"y=e%5Ex%2B%283-e%5E5%29\" for x greater than or equal to 5 [purple]

\n" ); document.write( "Here is a graph with the four functions overlapping, and showing the piecewise graph continuous at (1,1), (4,2), and (5,3):

\n" ); document.write( "\"graph%28400%2C400%2C-6%2C10%2C-6%2C10%2Cx%5E2%2Csqrt%28x%29%2Cx-2%2Ce%5Ex%2B3-e%5E5%29\"
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