Function Operations

Function Operations exercise

• Determine the total cost of the sand-proof glass carpet dome.

  Hints

  For example, if the production of one carpet dome costs $10$ dollars, with a one-time cost of $200$ dollars, then the total cost of producing $x$ carpets is $C_{\text{carpet}}(x)=10x+200$.

  To add functions together, just add the terms of both functions together:

  Solution

  The cost to make each carpet is 100 gold coins plus a one-time cost of 150 gold coins - to buy the loom used to weave the carpets.

  The total cost to produce the sand-proof glass dome is 50 gold coins per dome plus a one-time cost of 100 gold coins to buy the dome-building machinery.

  To figure out the total cost, let's write the costs of manufacturing as two different functions, and then add them together.

  Let $x$ represent the number of carpets produced and $C_{\text{carpet}}(x)$ represent the total cost of producing $x$ carpets. We then have that: $C_{\text{carpet}}(x)=100x+150$.

  Let $x$ represent the number of domes produced and $C_{\text{dome}}(x)$ represent the total cost of producing $x$ domes. We then have that: $C_{\text{dome}}(x)=50x+100$.

  If Jaanav produces $x$ carpets and domes, then we just have to add the total costs of the domes and the carpets together to get the total cost $C_{\text{total cost}}(x)$ of the sand-proof glass carpet dome!

  i.e. $C_{\text{total cost}}(x)=C_{\text{carpet}}(x)+C_{\text{dome}}(x)$.

  Specifically, $C_{\text{total cost}}(x)=100x+150+50x+100$. Combining like terms, we get $C_{\text{total cost}}(x)=150x+250$.

• Establish the equation to calculate Jaanav total profit.

  Hints

  The $750$ gold coins is a cost which needs to be paid from the money made from the carpet domes. So you have to subtract $750$ from the amount of money made.

  If one carpet dome costs $10$ dollars, then we get $10\times 10=100$ dollars for selling $10$ carpet domes.

  An example of combining like terms:

  $20x+30-15x=20x-15x+30=5x+30$

  Solution

  Let $x$ be the number of carpet domes Jaanav would like to sell and $C_{\text{initial profit}}(x)$ be the amount of gold coins Jaanav makes after covering the cost for his commercial. We then have that

  $C_{\text{initial profit}}(x)=250x-750$.

  To get the total profit $C_{\text{total profit}}(x)$ after selling $x$ carpet domes, we have to subtract the total cost from the initial profit:

  $C_{\text{total profit}}(x)=C_{\text{initial profit}}(x)-C_{\text{total cost}}(x)$.

  We can then conclude that the total profit is

  $C_{\text{total profit}}(x)=250x-750-(150x+250)=100x-1000$.

• Examine the total cost of the production with Janaav's brother's new glass dome machine.

  Hints

  Remember to combine like terms.

  To add terms with variables you just have to add the coefficients:

  Solution

  The total cost is the sum of the carpet cost and the dome cost.

  We have that the carpet cost is given by, $C_{\text{carpets}}(x)=100 x+ 150$, and that the dome cost is given by, $C_{\text{dome}}(x)=30x+150$.

  The sum of these functions gives us the total cost:

  $C_{\text{total cost}}(x)=(C_{\text{carpets}}+C_{\text{dome}})(x)=(100 x+ 150)+(30x+150)=100 x+ 150+30x+150$.

  Now we rearrange the terms so that the like terms are together,

  $C_{\text{total cost}}(x)=100x+30x+150+150$.

  And we combine like terms to get,

  $C_{\text{total cost}}(x)=130x+300$.

• Figure out the price for one dome.

  Hints

  The cost function is given by the item cost multiplied by the number of domes produced plus any one-time costs.

  The price which Janaav will break even at will give a profit of zero gold coins.

  You have to solve a linear equation with an unknown price $y$.

  Solution

  The break even point is when the amount of money made and the costs are the same.

  i.e. the total profit is zero.

  The function representing the costs is

  $C(x)=80x+200$,

  where the cost for advertising is included.

  Because the amount of money made for one dome is unknown, we represent it with the variable $y$. So we get the profit function,

  $P(x)=yx-(80x+200)$.

  We know that for $x=10$ the costs should be totally covered, that means that we want $P(10)=0$. This gives us the following equation with the unknown price $y$:

  $0=10y-(80(10)+200)$.

  We simplify the term inside the parentheses to $C(10)=80(10)+200=1000$ and solve the equation as follows:

  $\begin{array}{rcr} 0&=&10y-1000\\ +1000&&+1000\\ 1000&=&10y\\ \overline{\color{#669900}{\div 10}}&&\overline{\color{#669900}{\div 10}}\\ 100&=&y \end{array}$

  We then find that Janaav should charge 100 gold coins per dome.

• Decide what the correct rules are for combining functions.

  Hints

  An example of addition with $f(x)=2x+1$ and $g(x)=3x-2$:

  An example for division with $f(x)=2x+1$ and $g(x)=3x-2$:

  An example for multiplication with $f(x)=2x+1$ and $g(x)=3x-2$:

  Solution

  How can we add, subtract, multiply, or divide two functions?

  • $(f+g)(x)=f(x)+g(x)$
  • $(f-g)(x)=f(x)-g(x)$
  • $(f\times g)(x)=f(x)\times g(x)$
  • $(f\div g)(x)=f(x)\div g(x)$

  i.e. we add, subtract, multiply or divide the corresponding terms of the functions.

  For example, with $f(x)=2x+1$ and $g(x)=3x-2$:

  $(f-g)(x)=f(x)-g(x)=2x+1-(3x-2)=2x+1-3x+2=-x+3$.

• Calculate each operation with the given functions.

  Hints

  Remember to keep track of signs.

  Use the FOIL method for multiplying two binomials:

  • Multiply the First.
  • Next multiply the Outer.
  • Then the Inner.
  • Last multiply the Last.

  Solution

  To add, subtract, multiply, or divide functions, just add, subtract, multiply, or divide their terms:

  • $(f+g)(x)=f(x)+g(x)=12x+23+4x-3=12x+4x+23-3=16x+20$.

  • $(f-h)(x)=f(x)-h(x)=12x+23-(12x-9)=12x+23-12x+9=32$.

  • $(h\div g)(x)=\frac{h(x)}{g(x)}=\frac{12x-9}{4x-3}=\frac{3(4x-3)}{4x-3}=3$.

  • $(f\times h)(x)=f(x)\times h(x)=(12x+23)\times (4x-3)=48x^2+56x-69$.

  Note that for $(f\times h)(x)$ we use the FOIL method to help with our calculation.