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

Basic Functions & Graph Properties

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Basic Functions & Graph Properties

All basic functions have a known parent function & some properties describing the behaviour function.

What is a parent function?

A parent function is the simplest function of a family of functions that preserves the definition and shape of its entire family of functions.

Common Properties

  1. Odd Functions
  • f(−x)=−f(x)f(-x)=-f(x)
  • The function contains rotational (point) symmetry about the origin
  1. Even Functions
  • f(−x)=f(x)f(-x)=f(x)
  • The function is symmetric about the y-axis (ie:. line symmetry)
  1. Continuous Functions
  • A continuous function has no sudden changes in values.
  • There are no holes, asymptotes, & breaks on the graph
  • Can sketch the graph without having pen leaving the page
  1. Discontinuous Functions
  • A discontinuous function has sudden changes in values, known as discontinuities.
  • There are potential holes, asymptotes, and/or breaks on the graph
  • Cannot sketch the graph without the pen leaving the page

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  1. Increasing Functions
  • As the x-value increases over its domain (reading the graph from left to right), the y-value also increase
  1. Decreasing Functions
  • As the x-value increases over its domain (reading the graph from left to right), the y-value decreases
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Linear Functions

Parent Function:f(x)=xf(x)=x
  • Odd Function
  • Continuous on its domain
  • Increasing on(−∞,∞)(-\infin,\infin)
  • Domain: x∈Rx\in\mathbb{R}
  • Range: y∈Ry\in\mathbb{R}


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Quadratic Functions

Parent Function:f(x)=x2f(x)=x^2
  • Even function
  • Continuous on its domain
  • Increasing: (0,∞)(0,\infin)
  • Decreasing: (−∞,0)(-\infin,0)
  • Domain: (−∞,∞)(-\infin, \infin)
  • Range: [0,∞)[0,\infin)

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Cubic Functions

Parent Function:f(x)=x3f(x)=x^3
  • Odd function
  • Continuous on its domain
  • Increasing: (−∞,∞)(-\infin,\infin)
  • Domain: (−∞,∞)(-\infin, \infin)
  • Range: (−∞,∞)(-\infin, \infin)

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Reciprocal Functions

Parent Function:f(x)=1x     {x≠0}f(x)=\frac{1}{x}~~~~~\color{green}\footnotesize{\text{\{x}}\neq0\}
  • Odd function
  • Discontinuous on its domain
  • Decreasing: (0,∞)(0,\infin)
  • Domain: (0,∞)(0,\infin)
  • Range: (−∞,0) ∪(0,∞)(-\infin,0)~\cup(0,\infin)

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Radical Functions

Parent Function:f(x)=xf(x)=\sqrt{x}
  • Neither odd nor even
  • Continuous: (0,∞)(0,\infin)
  • Increasing: (0,∞)(0,\infin)
  • Domain: (0,∞)(0,\infin)
  • Range: [0,∞)[0,\infin)

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Exponential Functions

Parent Function:f(x)=(b)x     {b≠0}f(x)=(b)^x~~~~~\color{green}\footnotesize\{b\neq0\}
  • Neither even nor odd
  • Continuous on its domain
  • Increasing: (0,∞)(0,\infin)
  • Domain: (0,∞)(0,\infin)
  • Range: (0,∞)(0,\infin)

Example
f(x)=2xf(x)=2^x

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Logarithmic Functions

Parent Function:f(x)=logbx     {b≠0}f(x)=log_b{x}~~~~~\color{green}\footnotesize\{b\neq0\}
  • Neither even nor odd
  • Continuous on its domain
  • Increasing: (0,∞)(0,\infin)
  • Domain: (0,∞)(0,\infin)
  • Range: (−∞,∞)(-\infin,\infin)

Example 1
f(x)=log⁡2xf(x)=\log_2{x}



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Example: Basic Functions & Graph Properties

Determine the domain, range, intervals of increase and/or decrease for y=−x−5+1y=-\sqrt{x-5}+1 and state whether the function is continuous or discontinuous over its domain.

Sketch a graph of the function:

Domain: [5,∞)[5,\infin)
Range: (−∞,1](-\infin,1]
Decreasing over [5,∞)[5,\infin)
Continuous over its domain


Practice: Functions & Graph Properties

Determine the interval of increase/decrease for each of the following functions:
A.
Decreasing: (0,∞)(0,\infin)
B.
Decreasing: [−3,∞)[-3,\infin)
C.
Increasing: (−∞,0) ∪ (0,∞)(-\infin,0)~\cup~(0,\infin)
D.
Decreasing: (−∞,∞)(-\infin,\infin)
y=−1xy=-\frac{1}{x}
y=−x+3+1y=-\sqrt{x+3}+1
y=−(4)xy=-(4)^x
y=−log⁡3xy=-\log_3{x}

Practice: Functions & Graph Properties

True or False:
The function y=2x2+5x−3y=2x^2+5x-3 is even.

Practice: Functions & Graph Properties

The function f(x) is a quadratic function continuous over all real numbers. The function f(x) has the following properties:
  • Increases over the interval (−∞,4)(-\infin,4)
  • Decreases over the interval (4,∞)(4,\infin)
  • Contains a double root at 4
  • Is congruent to y=4x2y=4x^2
What is the equation of f(x)?
Extra Practice
Functions & Graph Properties
Practice: Functions & Graph Properties
The function f(x) is a quadratic function continuous over all real numbers. The function f(x) has the following properties:
Decreases over the interval (−∞,−38)\Bigg(-\infin,-\dfrac{3}{8}\Bigg)
Functions & Graph Properties
Practice: Functions & Graph Properties
The function f(x) is a quadratic function continuous over all real numbers. The function f(x) has the following properties:
Increases over the interval (−∞,−32)(-\infin,-32)
Functions & Graph Properties
Practice: Functions & Graph Properties
True or False:
The function y=−2x3+5y=-2x^3+5 is odd.
Functions & Graph Properties
Practice: Functions & Graph Properties
True or False:
The function y=xx2−1y=\dfrac{x}{x^2-1} is odd.
Functions & Graph Properties
Practice: Functions & Graph Properties
Determine the interval of increase/decrease for each of the following functions:
Functions & Graph Properties
Practice: Functions & Graph Properties
Determine the interval of increase/decrease for each of the following functions: