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Wize High School Grade 12 Pre-Calculus Textbook > Functions (Basics)

Operations with Functions

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Operations with Functions

Let f(x) f(x)~and g(x) g(x)~be two functions. There are 4 basic operations of functions that can be used:
  • Function Addition: f(x)+g(x)f(x)+g(x)
  • Function Subtraction: f(x)−g(x)f(x)-g(x)
  • Function Multiplication: f(x)⋅g(x)f(x)\cdot g(x)
  • Function Division: f(x)g(x);  g(x)≠0\frac{f(x)}{g(x)};~~g(x)\neq 0

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Example 1
Given f(x)=2x+1f(x)=2x+1 and g(x)=x2−5g(x)=x^2-5, find:
a. f(x)+g(x)f(x)+g(x)
f(x)+g(x)=(2x+1)+(x2−5)       =x2+2x−4f(x)+g(x)=(2x+1)+(x^2-5)~~~~~~~\newline{}\newline{} =x^2+2x-4

b. f(x)−g(x)f(x)-g(x)

f(x)−g(x)=(2x+1)−(x2−5)          =−x2+2x+6f(x)-g(x)=(2x+1)-(x^2-5)~~~~~~~~~~\newline{}\newline{} =-x^2+2x+6

c. f(x)⋅g(x)f(x)\cdot g(x)
f(x)⋅g(x)=(2x+1)(x2−5)                       =2x3−10x+x2−5=2x3+x2−10x−5f(x)\cdot g(x)=(2x+1)(x^2-5)~~~~~~~~~~~~~~~~~~~~~~~\newline{}\newline{} =2x^3-10x+x^2-5\newline{}\newline{} =2x^3+x^2-10x-5

d. f(x)g(x)\displaystyle\frac{f(x)}{g(x)}

f(x)g(x)=2x+1x2−5\frac{f(x)}{g(x)}=\frac{2x+1}{x^2-5}

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Function Composition (composite function)

Additionally, we can combine functions in one more by way composing them. (putting one function "inside" another)

To compute f(g(x))f\left(g\left(x\right)\right), substitute g(x)g\left(x\right) for xxin f(x)f\left(x\right). The domain of f(g(x))f\left(g\left(x\right)\right) is {x∈Dg:g(x)∈Df}\lbrace x\in D_g:g(x)\in D_f\rbrace.

Note: Another common notation for f(g(x))f\left(g\left(x\right)\right) is (f∘g)(x)\left(f\circ g\right)(x). ∘\circ is the symbol for composition.
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Example: Function Composition

Find (f∘g)(x)(f\circ g)(x) and its domain if

f(x)=1x−1\displaystyle f(x)=\frac{1}{x-1} and g(x)=1x+2\displaystyle g(x)=\frac{1}{x+2}

f∘g(x)=11x+2−1x+2≠0  ⟺  x≠−21x+2≠1  ⟺  x+2≠1  ⟺  x≠−1x≠−1,−2D:x∈(−∞,−2)∪(−2,−1)∪(−1,∞)\begin{array}{c} \displaystyle f\circ g(x)=\frac{1}{\frac{1}{x+2}-1} \\ \\ x+2\neq 0\iff x\neq -2\\ \\ \displaystyle \frac{1}{x+2}\ne 1 \iff x+2\neq1\iff x\neq -1 \\ \\x\neq -1,-2\\ \\D:x\in(-\infty,-2)\cup (-2,-1) \cup (-1,\infty) \end{array}

Example: Operations with Functions

Two functions, f(x) f(x)~in green and g(x) g(x)~in purple, and graphed below:

Find and graph the following:
  1. f(x)+g(x)f(x)+g(x)
  2. f(x)−g(x)f(x)-g(x)
  3. g(x)−f(x)g(x)-f(x)
  4. f(x)×g(x)f(x)\times g(x)
First, pull a table of values from the graph for f(x) and g(x) using the same x-coordinate points for both f(x) and g(x):
xf(x)xg(x)−44−49−33−34−22−21−11−10000111142229\begin{array} {|c|c||c|c|} \hline x&f(x)&x&g(x)\\\hline -4&4&-4&9\\ -3&3&-3&4\\ -2&2&-2&1\\ -1&1&-1&0\\ 0&0&0&1\\ 1&1&1&4\\ 2&2&2&9\\\hline \end{array}
a. f(x)+g(x):f(x)+g(x):

xf(x)g(x)f(x)+g(x)−44913−3347−2213−11010011114522911\begin{array} {|c|c|c|c|} \hline x&f(x)&g(x)&f(x)+g(x)\\\hline -4&4&9&13\\ -3&3&4&7\\ -2&2&1&3\\ -1&1&0&1\\ 0&0&1&1\\ 1&1&4&5\\ 2&2&9&11\\\hline \end{array}


b. f(x)−g(x):f(x)-g(x):

xf(x)g(x)f(x)−g(x)−449−5−334−1−2211−1101001−1114−3229−7\begin{array} {|c|c|c|c|} \hline x&f(x)&g(x)&f(x)-g(x)\\\hline -4&4&9&-5\\ -3&3&4&-1\\ -2&2&1&1\\ -1&1&0&1\\ 0&0&1&-1\\ 1&1&4&-3\\ 2&2&9&-7\\\hline \end{array}



c. g(x)−f(x):g(x)-f(x):

xf(x)g(x)g(x)−f(x)−4495−3341−221−1−110−1001111432297\begin{array} {|c|c|c|c|} \hline x&f(x)&g(x)&g(x)-f(x)\\\hline -4&4&9&5\\ -3&3&4&1\\ -2&2&1&-1\\ -1&1&0&-1\\ 0&0&1&1\\ 1&1&4&3\\ 2&2&9&7\\\hline \end{array}


d. f(x)×g(x):f(x)\times g(x):

xf(x)g(x)f(x)×g(x)−44936−33412−2212−11000010114422918\begin{array} {|c|c|c|c|} \hline x&f(x)&g(x)&f(x)\times g(x)\\\hline -4&4&9&36\\ -3&3&4&12\\ -2&2&1&2\\ -1&1&0&0\\ 0&0&1&0\\ 1&1&4&4\\ 2&2&9&18\\\hline \end{array}


The graphs of f(x)f(x) (in green) and g(x)g(x) (in purple) are provided on the same plot shown below:


Let f(x)=4x2−1 f(x)=4x^2-1~ and g(x)=2x−1g(x)=2x-1.

Determine f(x)g(x)\frac{f(x)}{g(x)} in simplified form.

Practice: Operations with Functions

Blake runs a tutoring company for high-school students. Blake has a fixed monthly cost of $400/month plus an additional cost of $5/student (for supplies, marketing, etc...) Blake charges $60/student.

Note: Let 'x' be the number of students
Extra Practice
Function Operations
Let f(x)=xx−3f(x)=\dfrac{x}{x-3} and let g(x)=22x+1g(x)=\dfrac{2}{2x+1}.
Determine the following, simplifying all answers (leave answer in factored form where possible):
Operations with Functions
Practice: Operations with Functions
George sells boxes of chocolates at the Ladner Market. It costs $550/month to run the stand plus an additional cost of $4.45/box for supplies. George sells a box of half-dozen chocolates for $8.95/box.
Note: Let 'x' be the number of boxes of chocolates sold.
Operations with Functions
Practice: Operations with Functions
Amy owns a skincare company. Her biggest seller is her "Rose oil Face Cream". Her fixed monthly costs are $1600/month plus an additional $12/jar for all supplies and ingredients. Amy charges $45/4oz jar of face cream.
Note: Let 'x' be the number of jars that will be sold.
Operations with Functions
Practice: Operations with Functions
Let f(x)=4x2+12x+9f(x)=4x^2+12x+9 and g(x)=4x2+4x−3g(x)=4x^2+4x-3.
Determine f(x)g(x)\dfrac{f(x)}{g(x)}.
Operations with Functions
Practice: Operations with Functions
Let f(x)=25x2+30x+25f(x)=25x^2+30x+25 and g(x)=5x+3g(x)=5x+3.
Determine f(x)g(x)\dfrac{f(x)}{g(x)}
Operations with Functions
Practice: Operations with Functions
The graphs of f(x)f(x) (in green) and g(x)g(x) (in blue) are provided on the same plot shown below:
Operations with Functions
Practice: Operations with Functions
The graphs of f(x)f(x) (in green) and g(x)g(x) (in blue) are provided on the same plot shown below: