Topics
1. Analyzing Functions
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Algebra 2

1.02 Function families

Lesson
Practice

Function families

There are many types of functions, and we can group them into categories called function families. Function families consist of a parent function and all transformations of the parent function. Let's look at the key features of several parent functions we will see in this course.

Square root function

A radical function with an index of 2. The square root parent function is represented by the form f\left(x\right)=\sqrt{x}, where x \geq 0.

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  • Domain: x \geq 0

  • Range: y \geq 0

  • Has an absolute minimum at \left(0,0\right)

  • x-intercept at \left(0,0\right)

  • y-intercept at \left(0,0\right)

  • Increasing over its domain

Cube root function

A radical function with an index of 3. The cube root parent function is represented by the form f\left(x\right)=\sqrt[3]{x}.

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  • Domain: all real numbers

  • Range: all real numbers

  • No absolute maximum or minimum

  • No relative maximum or minimum

  • x-intercept at \left(0,0\right)

  • y-intercept at \left(0,0\right)

  • Increasing over its domain

Rational function

A quotient of polynomials in which the denominator has a degree of at least 1. The rational parent functions are f\left(x\right)=\dfrac{1}{x} and f\left(x\right)=\dfrac{1}{x^2} where x \neq 0 and y \neq 0.

The rational parent function with a linear denominator is f\left(x\right)=\dfrac{1}{x}.

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  • Domain: x \neq 0

  • Range: y \neq 0

  • No absolute maximum or minimum

  • No relative maximum or minimum

  • No x-intercept or y-intercept

  • Asymptotes at the x-axis and y-axis

  • Decreasing over its domain

The rational parent function with a quadratic denominator is f\left(x\right)=\dfrac{1}{x^2}.

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  • Domain: x \neq 0

  • Range: y \neq 0

  • No absolute maximum or minimum

  • No relative maximum or minimum

  • No x-intercept or y-intercept

  • Asymptotes at the x-axis and y-axis

  • Increasing for x<0

  • Decreasing for x>0

Exponential function

A function where the independent variable is in the exponent. An exponential function can be written in the form f\left(x\right) = ab^x where a \neq 0 and b>0

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  • Domain: all real numbers

  • Range: y>0

  • Horizontal asymptote at the x-axis

  • Increasing when b>1

  • Decreasing when 0<b<1

  • No absolute maximum or minimum

  • No relative maximum or minimum

  • No x-intercept

  • y-intercept at \left(0,1\right)

Logarithmic function

A function f\left(x\right) that represents the exponent to which b must be raised to get x. The logarithmic parent function is represented by the form f\left(x\right)=\log_{b}x, where x > 0, b > 0, and b \neq 1.

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  • Domain: x > 0

  • Range: y \in \Reals

  • Increasing when b>1

  • Decreasing when 0 < b < 1

  • No absolute maximum or minimum

  • No relative maximum or minimum

  • x-intercept at (1,0)

  • No y-intercept

  • Asymptote at the y-axis

Absolute value function

A function that contains a variable expression inside absolute value bars; a function of the form f\left(x\right) = a\left|x - h\right| + k

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  • Domain: all real numbers

  • Range: y\geq 0

  • Has an absolute minimum at \left(0,0\right)

  • x-intercept at \left(0,0\right)

  • y-intercept at \left(0,0\right)

  • Has a vertical line of symmetry separating the increasing and decreasing intervals

We have previously studied other types of functions known as polynomial functions. Some examples of polynomial functions are shown.

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Constant function (degree =0)
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Linear function (degree =1)
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Quadratic function (degree =2)
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Polynomial function (degree >3)

Examples

Example 1

Identify the function family represented by each graph.

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Worked Solution
Create a strategy

Use the shape and characteristics of the graph to determine the function family represented. Characteristics to consider include: domain, range, x-intercept, y-intercept, absolute and relative minimums or maximums, and increasing/decreasing intervals.

Apply the idea

This graph represents a cube root function because it has a domain of all real numbers, has a range of all real numbers, and increases over its domain.

Reflect and check

Note that the graphs of transformed functions may not have the same exact key features as the parent function. This is because the key features of a function may change when it is transformed.

For example, the y-intercept of the parent cube root function is at \left(0,0\right), but the y-intercept of this cube root function is at \left(0,3\right).

Looking at the shape of the graph will be important for helping us determine the function family. Cube root functions will increase at a decreasing rate, then in the center, it changes and increases at an increasing rate. The point in the center is known as an inflection point, and we will learn more about this later.

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Worked Solution
Create a strategy

Use the shape and characteristics of the graph to determine the function family represented. Characteristics to consider include: domain, range, x-intercept, y-intercept, absolute and relative minimums or maximums, and increasing/decreasing intervals.

Apply the idea

This graph represents a rational function because it has no x or y-intercepts, and it is decreasing over its domain. Additionally, it appears to have both a vertical and a horizontal asymptote because its domain is x \neq 0 and its range is y \neq 1.

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Worked Solution
Create a strategy

Use the shape and characteristics of the graph to determine the function family represented. Characteristics to consider include: domain, range, x-intercept, y-intercept, absolute and relative minimums or maximums, and increasing/decreasing intervals.

Apply the idea

This graph represents a logarithmic function because it appears to have an asymptote at the y-axis, it has a domain of x>0 and a range of all real numbers, and it is increasing over its domain.

Reflect and check

Square root functions and logarithmic functions appear to have a similar shape. Both functions increase at an increasing rate over their domain.

The main difference between these functions is their range. A logarithmic function has a range of all real numbers, but a square root function's range does not include all real numbers.

Example 2

Determine the type of function represented by each table.

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x4581320
f\left(x\right)23456
Worked Solution
Create a strategy

To determine the type of function represented by a table, we can either plot the points on a coordinate plane and note the shape of the graph or examine how the function values increase or decrease.

Apply the idea
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Plotting and connecting the points on a graph, it appears the points represent a square root function.

  • Domain: x \geq 4

  • Range: y \geq 2

  • Has an absolute minimum at \left(4,2\right)

  • Increasing over its domain

  • No asymptotes

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x01234
f\left(x\right)812182740.5
Worked Solution
Create a strategy

We can begin by plotting the given points on the coordinate plane and observing the characteristics.

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It is difficult to identify characteristics with the limited domain provided by the table, and there could be a few options of functions this graph is representing. It could be:

  • A rational function
  • An exponential function
  • A polynomial function

From here, will we try to find a pattern in the way the y-values increase.

Apply the idea

An exponential function increases at a constant factor. Since the x-values increase by 1 each time, we can find the constant factor by dividing the outputs.\begin{aligned} 12\div8&=1.5\\18\div 12&=1.5\\27\div 18&=1.5\\40.5\div 27&=1.5 \end{aligned}Since the ratio of the outputs is the same, this function is exponential.

Example 3

Determine the type of function represented by each equation.

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g\left(x\right)=\dfrac{2}{x-1}

Worked Solution
Create a strategy

Use the structure of the equation to determine the function family represented. Notice that for the function families we have learned so far, the independent variable (usually x) is inside different symbols or located in different parts of the parent equations.

Apply the idea

Since the independent variable (x) is in the denominator, this is a rational function.

Reflect and check
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Graphing the equation can also help us determine what type of function is represented by the equation.

This graph appears to have a vertical and horizontal asymptote at y=0 and x=1, so it is easy to identify that it is rational.

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y=-2\sqrt{2x-4}

Worked Solution
Create a strategy

Use the structure of the equation to determine the function family represented.

Apply the idea

Since the independent variable (x) is located inside a radical with an index of 2, this is a square root function.

Idea summary

Function families consist of a parent function and all transformations of the parent function. The graphs of the parent functions within each family are shown.

Functions with their general equations and graphs shown below. From left to right, the functions are: Linear, f of x equals x; Quadratic, f of x equals x squared; Square root, f of x equals square root of x; Cubic, f of x equals x cubed; and Cube root, f of x equals cube root of x. Speak to your teacher for more details.
Functions with their general equations and graphs shown below. From left to right, the functions are: Absolute value, f of x equals absolute value of x; Rational-Linear, f of x equals 1 over x; Rational-Quadratic, f of x equals 1 over x squared; Exponential, f of x equals b raised to x where b is greater than 1; and Logarithmic, f of x equals log of x base b when b greater than 1. Speak to your teacher for more details.

Outcomes

A2.F.1

The student will investigate, analyze, and compare square root, cube root, rational, exponential, and logarithmic function families, algebraically and graphically, using transformations.

A2.F.1a

Distinguish between the graphs of parent functions for square root, cube root, rational, exponential, and logarithmic function families.

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