Introduction As is true with linear and exponential functions, we can perform operations on...

IntroductionAs is true with linear and exponential functions, we can perform operations on quadratic functions. Such operations include addition, subtraction, multiplication, and division. This lesson will focus on adding, subtracting, multiplying, and dividing functions to create new functions. The lesson will also explore the effects of dividing a quadratic by one of its linear factors.

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5.7.2: Operating on Functions

Key ConceptsOperations with Functions• Functions can be added, subtracted, multiplied, and

divided.

• For two functions f(x) and g(x), the addition of the functions is represented as follows:

• For two functions f(x) and g(x), the subtraction of the functions is represented as follows:

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Key Concepts, continued• For two functions f(x) and g(x), the multiplication of

the functions is represented as follows:

• For two functions f(x) and g(x), the division of the functions is represented as follows:

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Key Concepts, continued• Adding and subtracting linear expressions from a

quadratic will yield a quadratic.

• Multiplying and dividing a quadratic by anything other than a constant will not yield a quadratic.

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Key Concepts, continuedRestricted Domains • When considering the division of a quadratic by a

linear factor, it is possible to create a linear expression with a restricted domain. For example:

• For f(x) = x2 + 5x + 6 and g(x) = x + 3,

can be found such that

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Key Concepts, continued

• In simpler terms,

• Remember that the denominator of a fraction cannot equal 0.

• Set the denominator equal to 0 and solve for x to find the restricted value(s) in the domain: x + 3 = 0, so x ≠ –3.

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Key Concepts, continued• Given the similar function h(x) = x + 2, the domain is

all real numbers, and the range is the same. However,

since f(x) is divided by g(x), the domain of

from the preceding example is all real numbers except for x = –3 and the range is all real numbers except for y = –1.

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Key Concepts, continued• This is because when the restricted value of the domain

(–3) is substituted into the simplified form of

and solved for y, we get:

• Therefore, since x ≠ –3, then y ≠ –1.

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Common Errors/Misconceptions• forgetting to restrict the domain when dividing functions • not realizing that functions must be of the same

variable for like terms to be combined • having difficulty moving from the formal notation to a

workable problem where functions can be used with operations

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Guided Practice

Example 1Let f(x) = x2 – 3x + 4 and g(x) = x2 + 6x – 3. Build a new function, h(x), for which h(x) = (f + g)(x).

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Guided Practice: Example 1, continued

1. Expand the new function, h(x), into a form where substitution can be used.

h(x) = (f + g)(x) = f(x) + g(x)

The new function is expanded as h(x) = f(x) + g(x).

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2. Add the functions.

The new function is h(x) = 2x2 + 3x + 1.

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f(x) = x2 – 3x + 4 and g(x) = x2 + 6x – 3

Given functions from problem statement

h(x) = f(x) + g(x) Expanded notation

h(x) = (x2 – 3x + 4) + (x2 + 6x – 3)Substitute values for f(x) and g(x).

h(x) = 2x2 + 3x + 1 Combine like terms.

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Guided Practice

Example 3

For f(x) = 3x2 + 13x – 10 and g(x) = x + 5, find

What type of function is the quotient of Are

there restrictions on the domain and range of the

function

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1. Since the functions are being divided, write the functions f(x) and g(x) as a fraction.

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2. Factor the quadratic function, f(x).

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3. Simplify the equation and define the type of equation of the simplified form. Divide away the monomial (x + 5) from the top and bottom of the fraction:

The function is a linear equation that graph

the line y = 3x – 2.17



4. Look at the original fraction to see if there are restricted values on the domain. In this case, x ≠ –5 because (–5) + 5 = 0 and division by 0 is undefined. Next, substitute x = –5 into the final equation to determine the restricted value(s) of y.

3x – 2 = 3(–5) – 2 = –17

Since x ≠ –5, then y ≠ –17.

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Introduction As is true with linear and exponential functions, we can perform operations on...

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