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Analyze the pieces of the function.
Piecewise functions are written in the format:
behavioral piece,domain
behavioral piece,domain
behavioral piece,domain
⋮⋮
f of x equals a behavioral piece across a given domain, f of x equals another behavior piece across a given domain, f of x equals another behavioral piece across a given domain, etc.
Identify the function families that make up the piecewise function.
For example, given the piecewise function g(x)=
12x−2,x≤-negative2
2x,x>-negative2
For example, given the piecewise function g of x equals one half x minus 2 for x less than or equal to negative 2, and 2 to the power of x for x greater than negative 2
12x−2,x≤-negative2This piece is a linear function over the domain of all x values less than or equal to -negative2.
2x,x>-negative2This piece is an exponential function over the domain of all x values great than -negative2.
g of x equals one half x minus 2 for x less than or equal to negative 2. This function piece is a linear function over the domain of all x values less than or equal to -2. g of x equals 2 to the power of x for x greater than negative 2. This function piece is an exponential function over the domain of all x values greater than -2.
Graph the first piece of the function over the specified domain.
behavioral piece,domain Graph this piece over the specified domain.
behavioral piece,domain
behavioral piece,domain
⋮⋮
In a piecewise function, f of x equals a behavioral piece across a given domain, f of x equals another behavior piece across a given domain, f of x equals another behavioral piece across a given domain, etc. Graph the first behavioral piece over the specified domain.
For example,
12x−2,x≤-negative2This piece is a linear function over the domain of all x values less than or equal to -negative2.
2x,x>-negative2
g of x equals one half x minus 2 for x less than or equal to negative 2. This first function piece is a linear function over the domain of all x values less than or equal to -2. g of x equals 2 to the power of x for x greater than negative 2. This second function piece is an exponential function over the domain of all x values greater than -2.
A solid dot is needed at x=-negative2 since the domain of the function for this piece includes the function’s value for x=-negative2.
Graph the second piece of the function over the specified domain.
behavioral piece,domain
behavioral piece,domain Graph this piece over the specified domain.
behavioral piece,domain
⋮⋮
In a piecewise function, f of x equals a behavioral piece across a given domain, f of x equals another behavior piece across a given domain, f of x equals another behavioral piece across a given domain, etc. Graph the second behavioral piece over the specified domain.
For example,
12x−2,x≤-negative2This piece is a linear function over the domain of all x values less than or equal to -negative2.
2x,x>-negative2This piece is an exponential function over the domain of all x values great than -negative2.
g of x equals one half x minus 2 for x less than or equal to negative 2. This first function piece is a linear function over the domain of all x values less than or equal to -2. g of x equals 2 to the power of x for x greater than negative 2. This second function piece is an exponential function over the domain of all x values greater than -2.
An open dot is needed at x=-negative2 since the domain of the function for this piece includes all x values greater than x=-negative2, not including -negative2.
Continue to graph the pieces of the function over the specified domains.
behavioral piece,domain
behavioral piece,domain
behavioral piece,domain Graph this piece over the specified domain.
⋮⋮
In a piecewise function, f of x equals a behavioral piece across a given domain, f of x equals another behavior piece across a given domain, f of x equals another behavioral piece across a given domain, etc. Graph the third behavioral piece over the specified domain.
12x−2,x≤-negative2
2x,x>-negative2
g of x equals one half x minus 2 for x less than or equal to negative 2, and g of x equals 2 to the power of x for x greater than negative 2
only has two pieces, so the graph is complete.
12x−2,x≤-negative2
2x,x>-negative2
g of x equals one half x minus 2 for x less than or equal to negative 2, and g of x equals 2 to the power of x for x greater than negative 2
Remember, some piecewise functions can have many pieces, or intervals.