11. Modeling with Functions

Lesson

Just as we can perform operations on polynomials, so too can we perform operations on different functions - adding, subtracting, multiplying or dividing them, provided we follow specific rules. The table below defines each of the four operations for functions.

Operations with functions

We can perform operations with functions using the following rules:

Operation | Definition | Example using $f(x)=3x$f(x)=3x and $g(x)=2x-1$g(x)=2x−1 |
---|---|---|

Addition | $(f+g)(x)=f(x)+g(x)$(f+g)(x)=f(x)+g(x) |
$3x+2x-1=5x-1$3x+2x−1=5x−1 |

Subtraction | $(f-g)(x)=f(x)-g(x)$(f−g)(x)=f(x)−g(x) |
$3x-(2x-1)=x+1$3x−(2x−1)=x+1 |

Multiplication | $(f\cdot g)(x)=f(x)\cdot g(x)$(f·g)(x)=f(x)·g(x) |
$3x(2x-1)=6x^2-3x$3x(2x−1)=6x2−3x |

Division | $(\frac{f}{g})(x)=\frac{f(x)}{g(x)},g(x)\ne0$(fg)(x)=f(x)g(x),g(x)≠0 |
$\frac{3x}{2x-1},x\ne\frac{1}{2}$3x2x−1,x≠12 |

Two functions $f$`f` and $g$`g` may be combined as a sum $f+g$`f`+`g`, meaning that for each $x$`x` in the common domain we add the function values $f(x)$`f`(`x`) and $g(x)$`g`(`x`) to get $(f+g)(x)$(`f`+`g`)(`x`). The same can be done with the difference of two functions.

Note that this operation makes no sense unless $x$`x` belongs to the domains of both $f$`f` and $g$`g`. It may be necessary to restrict the domain of one or both functions to meet this requirement.

Let $f(x)=x^2$`f`(`x`)=`x`2 and $g(x)=2x+1$`g`(`x`)=2`x`+1.

**Find** $(f+g)(x)$(`f`+`g`)(`x`) and its domain.

**Think:** The domains of both functions are the real numbers. So, the sum function $(f+g)(x)$(`f`+`g`)(`x`) will also have the real numbers for its domain.

**Do:** We have, $(f+g)(x)=f(x)+g(x)$(`f`+`g`)(`x`)=`f`(`x`)+`g`(`x`) for each $x$`x` in the domain. Therefore, $(f+g)(x)=x^2+2x+1$(`f`+`g`)(`x`)=`x`2+2`x`+1, and Its domain is the set of real numbers.

We combine functions $f$`f` and $g$`g` as a product $fg$`f``g` by defining $(fg)(x)=f(x)\cdot g(x)$(`f``g`)(`x`)=`f`(`x`)·`g`(`x`) for each $x$`x` in the common domain of $f$`f` and $g$`g`.

Let $f(x)=\frac{3}{x}$`f`(`x`)=3`x` and $g(x)=2x-\frac{1}{3}$`g`(`x`)=2`x`−13.

**Find** $(fg)(x)$(`f``g`)(`x`) and its domain.

**Think:** The product function $(fg)(x)$(`f``g`)(`x`) is given by $f(x)\cdot g(x)$`f`(`x`)·`g`(`x`) over the domain $\left\{R|x\ne0\right\}${`R`|`x`≠0}. The domain has to be restricted to the real numbers without zero because this is the domain of $f$`f`.

**Do:** Therefore, $(fg)(x)=\frac{3}{x}\cdot\left(2x-\frac{1}{3}\right)$(`f``g`)(`x`)=3`x`·(2`x`−13) and so,

$(fg)(x)=6-\frac{1}{x}$(`f``g`)(`x`)=6−1`x`

We can define a quotient function $h(x)=\frac{f(x)}{g(x)}$`h`(`x`)=`f`(`x`)`g`(`x`) in a similar way to the way we defined the other operations, provided the domains of $f$`f` and $g$`g` are the same and we do not include in the domain values of $x$`x` that make $g(x)=0$`g`(`x`)=0.

Such functions are called **rational functions** when $f$`f` and $g$`g` are both polynomials.

Given $f(x)=x^2+5x+6$`f`(`x`)=`x`2+5`x`+6 and $g(x)=2x+1$`g`(`x`)=2`x`+1,

**Find** $(\frac{f}{g})(x)$(`f``g`)(`x`) and its domain.

**Think:** The quotient function is given by $(\frac{f}{g})(x)=\frac{f(x)}{g(x)}$(`f``g`)(`x`)=`f`(`x`)`g`(`x`), provided that $g(x)\ne0$`g`(`x`)≠0. First, we must find the quotient. Then determine restrictions on its domain by finding the values that make the denominator zero.

**Do:** Therefore, $(\frac{f}{g})(x)=\frac{x^2+5x+6}{2x+1}$(`f``g`)(`x`)=`x`2+5`x`+62`x`+1, provided that $x\ne\frac{1}{2}$`x`≠12.

If $f(x)=3x-5$`f`(`x`)=3`x`−5 and $g(x)=5x+7$`g`(`x`)=5`x`+7, find each of the following:

$(f+g)(x)$(

`f`+`g`)(`x`)$(f+g)$(

`f`+`g`)$\left(4\right)$(4)$(f-g)(x)$(

`f`−`g`)(`x`)$(f-g)$(

`f`−`g`)$\left(10\right)$(10)

Let $f\left(x\right)=8x^3+27$`f`(`x`)=8`x`3+27 and $g\left(x\right)=2x+3$`g`(`x`)=2`x`+3.

What is the domain of $(f/g)(x)$(

`f`/`g`)(`x`)? Give your answer in interval notation.Simplify the function $(f/g)(x)$(

`f`/`g`)(`x`):

The financial team at The Gamgee Cooperative wants to calculate the profit, $P\left(x\right)$`P`(`x`), generated by producing $x$`x` units of wetsuits.

The revenue produced by the product is given by the equation is $R\left(x\right)=-\frac{x^2}{4}+40x$`R`(`x`)=−`x`24+40`x`. The cost of production is given by the equation $C\left(x\right)=5x+410$`C`(`x`)=5`x`+410.

The profit is calculated as $P\left(x\right)=R\left(x\right)-C\left(x\right)$`P`(`x`)=`R`(`x`)−`C`(`x`).

Find an equation for $P\left(x\right)$

`P`(`x`).Find the values of the following:

$R\left(70\right)$

`R`(70)$=$=$\editable{}$$C\left(70\right)$

`C`(70)$=$=$\editable{}$$P\left(70\right)$

`P`(70)$=$=$\editable{}$Which of the following correctly displays the graphs of $y=R\left(x\right)$

`y`=`R`(`x`), $y=C\left(x\right)$`y`=`C`(`x`) and $y=P\left(x\right)$`y`=`P`(`x`)?The graph of $P\left(x\right)$

`P`(`x`) is represented by the black line in each option.Loading Graph...ALoading Graph...BLoading Graph...CLoading Graph...DLoading Graph...ALoading Graph...BLoading Graph...CLoading Graph...D

Write a function that describes a relationship between two quantities.

Combine standard function types using arithmetic operations.