Piecewise function

Question

This is a piecewise function....I'm not understanding my notes in class...can someone explain how to solve this please?

f(x)= 1/4x+2 if x < -2

-(x-1)^2+1 if -2≤x≤2

               2 if x>2

solve for f(-1) f(2) f(4)

Answer 1

Math if full of definitions, formulas, and procedures.  LEARN THEM ALL.

 

The definition of a function is "a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output" (Wikipedia).

 

Since this function may not be easily written as a single expression (like a line or a curve often can), the line segments and points are written with a range (or value) specified for x.  There is still only one y for each x.

 

Piecemeal means a little bit at a time.

 

So, for values of x less than -2, there is one function, for values of x between -2 and 2 (inclusive) there is another function, and for values greater than 2, there is a third.

 

Given any value for x, we use that criteria to determine which function to use for calculating the unique value of y.

 

Piecemeal functions appear in such things as electric bills, taxes and salaries.  For electric bills, you pay one rate for usage up to a certain value, then another rate for extra usage.  For taxes, you pay one rate until you get to the higher bracket, then you pay a higher rate.  For salaries, you get paid regular hourly rate up to 40 hours, then you get overtime (a higher rate) for all hours worked over 40.

 

In your problem, you are asked to find the function value y(x) for values of x of -1 and for x = 2 and for x = 4.

 

You can do that!  Go for it!

Answer 2

Since -1 is in the interval -2 ≤ x ≤ 2 , f(-1) = -(-1-1)² + 1 = -3

 

Since 2 is in the interval -2 ≤ x ≤ 2, f(2) = -(2-1)² + 1 = 0

 

Since 4 is greater than 2, f(4) = 2 

Answer 3

Evaluate the function at the appropriate pieces...

 

When x=-1...we are in the first piece:

 

f(-1)=1/4(-1)+2

 

When x=2 use the second piece:

 

f(2)=-(x-1)^2+1

 

and when x>2 the function is constant at f(x)=2, so f(4)=2.

Answer 4

A piecewise function is a function that has two parts.  Each part has its own domain.

 

 

I assume we are multiply these functions.

 

f(-1) * f(2) * f(4) =

 

[-(-1 - 1)² + 1] * [-(2 - 1)² + 1] * 2 =

 

[-(-2)² + 1] * [-2(1)² + 1] * 2 =

 

(-3)(-1)(2) =

 

6