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Wize High School Grade 12 Pre-Calculus Textbook > Combination of Functions

Product & Quotient of Functions

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Product & Quotient of Functions

Functions can be combined through multiplication and division.


Let f(x)f(x) and g(x)g(x) be two continuous functions defined on the interval (−∞,∞)(-\infin,\infin).
  • The product of 2 functions, f(x)f(x) and g(x)g(x), can be expressed as f(x)⋅g(x)\boxed{f(x)\cdot{}g(x)}.
  • The quotient of 2 functions, f(x)f(x) and g(x)g(x), can be expressed as f(x)g(x)\boxed{\dfrac{f(x)}{g(x)}}.
Wize Tip
Both functions must be defined at a point for the combination to be defined and g(x)≠0\colorThree{g(x)\neq{}0}.

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Example

Let f(x)=x2−4f(x)=x^2-4 and g(x)=x−2g(x)=x-2. Find:
  1. f(x)⋅g(x)f(x)\cdot{}g(x)
  2. f(x)g(x)\dfrac{f(x)}{g(x)}
1.f(x)⋅g(x)=(x2−4)⋅(x−2)=x3−2x2−4x+82.f(x)g(x)=(x2−4)(x−2)=(x−2)(x+2)(x−2)=x+2\begin{array}{rrcl} 1.&f(x)\cdot{}g(x)&=&(x^2-4)\cdot(x-2)\\\\ &&=&x^3-2x^2-4x+8\\\\\\ 2.&\dfrac{f(x)}{g(x)}&=&\dfrac{(x^2-4)}{(x-2)}\\\\ &&=&\dfrac{(x-2)(x+2)}{(x-2)}\\\\ &&=&x+2 \end{array}

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Example: Product & Quotient of Functions

If f(x)=x+4f(x)=\sqrt{x+4} and g(x)=x2−1g(x)=x^2-1, determine:
  1. f(x)⋅g(x)f(x)\cdot{}g(x)
  2. f(x)g(x)\dfrac{f(x)}{g(x)}
  3. Algebraically
  4. Graphically
State the domain for both 1 & 2.


1. f(x)⋅g(x)f(x)\cdot{}g(x)

a. f(x)⋅g(x)=(x+4)(x2−1)=(x2−1)x+4\begin{array}{rcl} f(x)\cdot{}g(x)&=&(\sqrt{x+4})(x^2-1)\\\\ &=&(x^2-1)\sqrt{x+4} \end{array}

The domain is x≥4x\geq{}4.


b.
xf(x)g(x)f(x)⋅g(x)−40150−3188−22332−130002−12150026336378874815302\begin{array}{c|c|c|c} x&f(x)&g(x)&f(x)\cdot{}g(x)\\\hline -4&0&15&0\\\hline -3&1&8&8\\\hline -2&\sqrt{2}&3&3\sqrt{2}\\\hline -1&\sqrt{3}&0&0\\\hline 0&2&-1&2\\\hline 1&\sqrt{5}&0&0\\\hline 2&\sqrt{6}&3&3\sqrt{6}\\\hline 3&\sqrt{7}&8&8\sqrt{7}\\\hline 4&\sqrt{8}&15&30\sqrt{2}\\ \end{array}


The graph of f(x)g(x)f(x)g(x):


The domain is x≥4x\geq{}4.


2. f(x)g(x)\dfrac{f(x)}{g(x)}

f(x)g(x)=x+4(x2−1)\begin{array}{rcl} \dfrac{f(x)}{g(x)}&=&\dfrac{\sqrt{x+4}}{(x^2-1)}\\\\ \end{array}

The domain is x≥−4, x≠±1, or in interval notation: [−4,−1)∪(−1,∞)x\ge-4,~x\ne\pm1,\ \text{or in interval notation:}\ [-4,-1)\cup(-1,\infin)

xf(x)g(x)f(x)g(x)−40150−3181/8−22332−130002−12150026336378874815302\begin{array}{c|c|c|c} x&f(x)&g(x)&\dfrac{f(x)}{g(x)}\\\hline -4&0&15&0\\\hline -3&1&8&1/8\\\hline -2&\sqrt{2}&3&3\sqrt{2}\\\hline -1&\sqrt{3}&0&0\\\hline 0&2&-1&2\\\hline 1&\sqrt{5}&0&0\\\hline 2&\sqrt{6}&3&3\sqrt{6}\\\hline 3&\sqrt{7}&8&8\sqrt{7}\\\hline 4&\sqrt{8}&15&30\sqrt{2}\\ \end{array}

The graph of f(x)g(x)\dfrac{f(x)}{g(x)}:


The domain is x≥−4, x≠±1x\geq{-4},~x\neq\pm1.

Practice: Product & Quotient of Functions

Let f(x)=16−x2f(x)=\sqrt{16-x^2} and g(x)=2x+1g(x)=2x+1. Determine:

Practice: Product & Quotient of Functions

If f(x)=x+2f(x)=x+2 and g(x)=x−3g(x)=x-3, then sketch a graph of y=f(x)g(x)y=\dfrac{f(x)}{g(x)} and determine the domain.

Practice: Product & Quotient of Functions


Determine the domain of f(x)g(x)f(x)g(x) if f(x)=log⁡2xf(x)=\log_{2}x and g(x)=xg(x)=\sqrt{x}.

Extra Practice
Product & Quotient of Functions
Practice: Product & Quotient of Functions
Determine the domain of f(x)g(x)f(x)g(x) if f(x)=25−4x2f(x)=\sqrt{25-4x^2} and g(x)=∣x−3∣−4g(x)=|x-3|-4. Express answer in interval notation.
Product & Quotient of Functions
Practice: Product & Quotient of Functions
Determine the domain of f(x)g(x)f(x)g(x) if f(x)=log⁡2(x2−3)f(x)=\log_{2}{(x^2-3)} and g(x)=x3g(x)=\sqrt{x^3}.
Product & Quotient of Functions
Practice: Product & Quotient of Functions
If f(x)=4x2+4x+1f(x)=\sqrt{4x^2+4x+1} and g(x)=xg(x)=x, then sketch a graph of y=f(x)g(x)y=f(x)g(x) and determine the domain.
Product & Quotient of Functions
Practice: Product & Quotient of Functions
If f(x)=x−1f(x)=x-1 and g(x)=2x+3g(x)=2x+3, then sketch a graph of y=f(x)g(x)y=\dfrac{f(x)}{g(x)} and determine the domain.
Product & Quotient of Functions
Practice: Product & Quotient of Functions
Let f(x)=log⁡(x+1)f(x)=\log(x+1) and g(x)=x2g(x)=x^2. Determine:
Product & Quotient of Functions
Practice: Product & Quotient of Functions
Let f(x)=x2+2x+1f(x)=\sqrt{x^2+2x+1} and g(x)=x−3g(x)=x-3. Determine:
Product & Quotient of Functions
If f(x)=2x+1f(x)=2x+1 and g(x)=x+4g(x)=x+4, sketch a graph of f(x)g(x)\dfrac{f(x)}{g(x)}.