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Wize University Calculus 1 Textbook > Pre-Calculus (Review)

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}
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.
Extra Practice
Domain & Range
Practice: Domain & Range
If f(x)=2−xf\left(x\right)=2-\sqrt{x} and g(x)=x2−9g\left(x\right)=x^2-9, find the domain and range of f(g(x))f\left(g\left(x\right)\right).
Function composition
Consider the functions f(x)=x1/23−xf(x)=\frac{x^{1/2}}{\sqrt{3-\sqrt{x}}} and g(x)=(6−x)2g(x)=(6-x)^2. Find (f∘g)(x)(f\circ g)(x) and the domain.
Domain & Range
Practice: Domain & Range
If f(x)=2−xf\left(x\right)=2-\sqrt{x} and g(x)=x2−9g\left(x\right)=x^2-9, find the domain and range of f(g(x))f\left(g\left(x\right)\right).
Q:\bf{Q:} Find f(g(x))f(g(x)) and its domain if f(x)=1x−1f\left(x\right)=\dfrac{1}{x-1} and g(x)=1x+2g\left(x\right)=\dfrac{1}{x+2}
Let f(x)f(x) and g(x)g(x) be two functions with values given in the following table:
The value of f(g(2))+g(f(2))f\left(g\left(2\right)\right)+g\left(f\left(2\right)\right) is:
Q:\bf{Q:} State the possible values of ff and gg given f(g(x))=x−ln⁡xf\left(g\left(x\right)\right)=\sqrt{x-\ln x}
Domain & Range
Practice: Domain & Range
If f(x)=2−xf\left(x\right)=2-\sqrt{x} and g(x)=x2−9g\left(x\right)=x^2-9, find the domain and range of f(g(x))f\left(g\left(x\right)\right).
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:
Domain & Range
Practice: Domain & Range
If f(x)=2−xf\left(x\right)=2-\sqrt{x} and g(x)=x2−9g\left(x\right)=x^2-9, find the domain and range of f(g(x))f\left(g\left(x\right)\right).
Domain & Range
Practice: Domain & Range
If f(x)=2−xf\left(x\right)=2-\sqrt{x} and g(x)=x2−9g\left(x\right)=x^2-9, find the domain and range of f(g(x))f\left(g\left(x\right)\right).
Let f(x)f(x) and g(x)g(x) be two functions with values given in the following table:
The value of f(g(2))+g(f(2))f\left(g\left(2\right)\right)+g\left(f\left(2\right)\right) is:
Q:\bf{Q:} Find f(g(x))f(g(x)) and its domain if f(x)=1x−1f\left(x\right)=\dfrac{1}{x-1} and g(x)=1x+2g\left(x\right)=\dfrac{1}{x+2}
Q:\bf{Q:} State the possible values of ff and gg given f(g(x))=x−ln⁡xf\left(g\left(x\right)\right)=\sqrt{x-\ln x}
Function composition
Consider the functions f(x)=x1/23−xf(x)=\frac{x^{1/2}}{\sqrt{3-\sqrt{x}}} and g(x)=(6−x)2g(x)=(6-x)^2. Find (f∘g)(x)(f\circ g)(x) and the domain.
Operations with Functions
Consider the functionsf(x)=x23−x2f(x)=\frac{x^2}{\sqrt{3-x^2}} and g(x)=6−xg(x)=\sqrt{6-x}. Find the domain of (f∘g)(x)(f \circ g)(x).
Practice: Functions Properties
Practice: Function Properties
If f(x)=∣x−2∣, g(x)=3−x+1f\left(x\right)=\left|x-2\right|,\ g\left(x\right)=3-\sqrt{x+1}, which of the following is true about h(x)=g(f(x))h\left(x\right)=g\left(f\left(x\right)\right)?
Domain & Range
Practice: Domain & Range
If f(x)=2−xf\left(x\right)=2-\sqrt{x} and g(x)=x2−9g\left(x\right)=x^2-9, find the domain and range of f(g(x))f\left(g\left(x\right)\right).