Function Basics

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1. Sketch f(x)=3x+2
   1. Domain?
      Solution
      x\in\mathbb{R}
      
      
   2. Range?
      Solution
      y\in\mathbb{R}
      
      
2. f(x)=3x-1, x\geq2
   1. Domain?
      Solution
      x\geq2
      
      
   2. Range?
      Solution
      y\geq5
      
      
   3. Evaluate f(4)
      Solution
      11
      
      
3. Sketch y=1-x, x\in\mathbb{Z}
   Solution
   
   
   
   
4. What is the range of f(x)=3?
   Solution
   y\in\{3\} or y=3
   
   
5. Write the domain notation x\geq3 using:
   1. brackets
      Solution
      [3,\infty)
   2. a number line
      Solution
      
      
      
6. Write the range notation y<-2 using:
   1. brackets
      Solution
      (-\infty,-2)
   2. a number line
      Solution
      
      
      
7. Write the domain notation for -1\leq x\leq3 using:
   1. brackets
      Solution
      [-1,3]
   2. a number line
      Solution
      
      
      
      
8. Write the range notation for -3\leq y<2 using:
   1. brackets
      Solution
      [-3,2)
   2. a number line
      Solution
      
      
      
9. Find the domain and range of the following graphs:
   1. See below:
      
      Solution
      x\in\mathbb{R} and y\leq4
      
      
   2. See below:
      
      Solution
      -4\leq x\leq 4
      
      
   3. See below:
      
      Solution
      x\geq 4 and y\geq 0
      
      
   4. See line below:
      
      Solution
      x\neq 3 and y\neq3
      
      
   1. Challenge – See below:
      
      Solution
      x\geq0 and y\leq9
      
      
10. Is the following graph a function?
    1. See below:
       
       Solution
       Yes
       
       
    2. See below:
       
       Solution
       Yes
       
       
    3. See below:
       
       Solution
       No
       
       
    4. See below:
       
       Solution
       Yes (although the function is undefined at x=0
       
       
    5. See below:
       
       Solution
       Yes
       
       
    6. See below:
       
       Solution
       Yes
       
       
    7. See below:
       
       Solution
       No
       
       
    8. See below:
       
       Solution
       No
       
       
    9. See below:
       
       Solution
       Yes
       
       
    10. See below:
        
        Solution
        No
        
        
11. Sketch both of the following lines on the same graph:
    y_1=x and y_2=-x+2
    Solution
    
    
    
    
12. Instead of using subscripts, letters can be more descriptive. Fiona’s wealth is modelled by f(x)=2x+10
    Gary’s wealth is modelled by g(x)=3x+5
    Henry’s wealth is modelled by -x+20
    Draw y=f(x), y=g(x), and y=h(x) all on the same graph.
    Solution
    
    
    
    
13. Money is a function of time in hours. Given M(t)=50t+200.
    1. How much does the plumber charge for just showing up?
       Solution
       $200
       
       
    2. How much does the plumber charge for working 8 hours?
       Solution
       $600
       
       
    3. How long does the plumber need to work to earn $400?
       Solution
       4 hours
       
       
    4. If possible, evaluate M(-4)
       Solution
       In the context of this question, there is no such thing as working negative 4 hours so there is no solution
       
       
14. The volume of a sphere is a function of its radius
    Given V(r)=\frac{4}{3}\pi r^3, evaluate V(2)
    Solution
    \frac{32}{3}\pi units cubed