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1) Let  \(f(x) = \left\{ \begin{array}{cl} 3x & \text{if } x < 3, \\ 3^x & \text{if } x \ge 3. \end{array} \right.\) Find \(f(2) + f(3) + f(4).\) 

2) Let \(f(x) = \left\{ \begin{array}{cl} -2x & \text{if } x < 0, \\ \frac{x}{2} & \text{if } x \ge 0. \end{array} \right.\) Find the range of f(x) in interval notation.

3) Find the area of the region that lies below the graph of y = 3 - |x - 1| but above the -axis.

 May 18, 2019
 #1
avatar+8720 
+2

1)

 

\(f(x) = \left\{ \begin{array}{cl} 3x & \text{if } x < 3, \\ 3^x & \text{if } x \ge 3. \end{array} \right.\)

 

 

Let's find  f(2)

 

2  <  3     so  we use   f(x)  =  3x

 

f(2)  =  3(2)

 

f(2)  =  6

 

Let's find  f(3)

 

3  ≥  3     so we use   f(x)  =  3x

 

f(3)  =  33

 

f(3)  =  27

 

Let's find  f(4)

 

4  ≥  3  so we use   f(x)  =  3x

 

f(4)  =  . . .?  Can you figure this one out?

 May 18, 2019
 #2
avatar+8720 
+3

2)
\(f(x) = \left\{ \begin{array}{cl} -2x & \text{if } x < 0, \\ \frac{x}{2} & \text{if } x \ge 0. \end{array} \right.\)

 

Let  y  =  f(x)  so we can say that...

 

The range includes the smallest possible y value to the biggest possible y value.
 

A graph might help:

 

The smallest possible y value is  0

 

There isn't a biggest possible y value because we can always find a bigger one, so we say it's  ∞

 

So the range is  [0, ∞)

 May 18, 2019

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