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Wize High School Grade 11 Math Textbook > Introduction to Functions

Properties of Parent Functions

Function NotationPrevious Section
Determining Domain & RangeNext Section
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Properties of Parent Functions

A parent function is the most basic form of some common functions. Let's take a closer look at their properties.

Linear

  • The linear function f(x)=xf(x)=x looks like a straight line through the origin. It has a slope of 1.
  • Domain: all real numbers -- x ∈Rx\ \in\reals
  • Range: all real numbers -- y∈Ry \in \reals


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Quadratic

  • The quadratic function f(x)=x2f(x)=x^2 has a distinctive U-shape with its vertex at the origin.
  • Domain: all real numbers -- x ∈Rx\ \in\reals
  • Range: non-negative real numbers -- {y∈R ∣ y≥0}\{ y \in \reals\ |\ y \ge 0 \}


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Square Root

  • The square root function f(x)=xf(x)=\sqrt x increases from left to right, but the slope gets shallower.
  • Domain: non-negative real numbers -- {x∈R ∣ x≥0}\{ x \in \reals\ |\ x \ge 0 \}
  • Range: non-negative real numbers -- {y∈R ∣ y≥0}\{ y \in \reals\ |\ y \ge 0 \}



Wize Tip
The value of the square root is defined only for positive numbers.
That's why there is no graph on the left side where xx is negative!

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Reciprocal

  • The reciprocal function f(x)=1xf(x) = \dfrac{1}{x} is split into two parts: one in quadrant 1, and another in quadrant 3.
  • Domain: all real numbers except 0 -- {x∈R ∣ x≠0}\{ x \in \reals\ |\ x \ne 0 \}
  • Range: all real numbers except 0 -- {y∈R ∣ y≠0}\{ y \in \reals\ |\ y \ne 0 \}




Wize Tip
Since we can not divide by zero, there is no value on the graph when x=0x=0, and we call this a vertical asymptote.

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Absolute Value

Wize Concept
The absolute value works by taking in a number and making it positive.
Ex. If f(x)=∣x∣f(x) = |x|, then f(−4.1)=∣−4.1∣=4.1f(-4.1)=|-4.1| = 4.1
  • The absolute value function f(x)=∣x∣f(x) = |x| has a distinctive V-shape with a cusp (corner) at the origin.
  • Domain: all real numbers -- x∈Rx \in \reals
  • Range: non-negative real numbers -- {y∈R ∣ y≥0}\{ y \in \reals\ |\ y \ge 0 \}


Practice: Basic Graphs

Match the graph with its function.
A.
B.
C.
D.
E.
f(x)=xf(x)=x
f(x)=x2f(x)=x^2
f(x)=∣x∣f(x)=|x|
f(x)=xf(x)=\sqrt{x}
f(x)=1xf(x)=\dfrac{1}{x}

Practice: Domain and Range of Basic Functions


Match each function with its domain and range.
A.
f(x)=xf(x)=\sqrt{x}
B.
f(x)=x2f(x)=x^2
C.
f(x)=xf(x)=x
D.
f(x)=1xf(x)=\dfrac{1}{x}
Domain: All real numbers
Range: All real numbers
Domain: All real numbers
Range: [0,∞)[0, \infty)
Domain: [0,∞)[0, \infty)
Range: [0,∞)[0, \infty)
Domain: (−∞,0)⋃(0,∞)(-\infty, 0) \bigcup (0, \infty)
Range: (−∞,0)⋃(0,∞)(-\infty, 0) \bigcup (0, \infty)

Practice: Basic Graphs

Of the five parent functions covered, which one does not have a value when x=0x=0?

Practice: Basic Graphs

Which of the five parent functions goes through the point (−1,1)(-1,1)? [Select all that apply]

Practice: Basic Graphs


The following graphs are related to the basic equations. Look at their distinctive features and match them with the family of equations they are related to.
A.
B.
C.
D.
Absolute Value
Quadratic
Square Root
Reciprocal

Basic Graphs

This reference sheet is super helpful for this course. Make sure to either create or print a copy of it to keep with you.